Volume
LIPIcs, Volume 264
38th Computational Complexity Conference (CCC 2023)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC)
Event
CCC 2023, July 17-20, 2023, Warwick, UKEditor
Publication Details
- published at: 2023-07-10
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-282-2
- DBLP: db/conf/coco/coco2023
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Complete Volume
LIPIcs, Volume 264, CCC 2023, Complete Volume
Abstract
LIPIcs, Volume 264, CCC 2023, Complete Volume
Cite as
38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 1-936, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@Proceedings{tashma:LIPIcs.CCC.2023,
title = {{LIPIcs, Volume 264, CCC 2023, Complete Volume}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {1--936},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023},
URN = {urn:nbn:de:0030-drops-182690},
doi = {10.4230/LIPIcs.CCC.2023},
annote = {Keywords: LIPIcs, Volume 264, CCC 2023, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 0:i-0:xiv, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{tashma:LIPIcs.CCC.2023.0,
author = {Ta-Shma, Amnon},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {0:i--0:xiv},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.0},
URN = {urn:nbn:de:0030-drops-182703},
doi = {10.4230/LIPIcs.CCC.2023.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Separation of the Factorization Norm and Randomized Communication Complexity
Abstract
In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds.
We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n).
As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23).
Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.
Cite as
Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, and Morgan Shirley. Separation of the Factorization Norm and Randomized Communication Complexity. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{cheung_et_al:LIPIcs.CCC.2023.1,
author = {Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Shirley, Morgan},
title = {{Separation of the Factorization Norm and Randomized Communication Complexity}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {1:1--1:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.1},
URN = {urn:nbn:de:0030-drops-182714},
doi = {10.4230/LIPIcs.CCC.2023.1},
annote = {Keywords: Factorization norms, randomized communication complexity}
}
Document
Border Complexity of Symbolic Determinant Under Rank One Restriction
Abstract
VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where the size of each A_i is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation i.e. whether VBP = VBP^ ̅. The power of approximation is well understood for some restricted models of computation, e.g. the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Markus Bläser et al., 2020], whereas the approximative closure of the last one captures the entire class of polynomial families computable by polynomial-sized formulas [Bringmann et al., 2017].
In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where for each 1 ≤ i ≤ n, A_i is of rank one. This class has been studied extensively [Edmonds, 1968; Jack Edmonds, 1979; Murota, 1993] and efficient identity testing algorithms are known for it [Lovász, 1989; Rohit Gurjar and Thomas Thierauf, 2020]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Plücker embedding of) a Grassmannian variety is closed.
Cite as
Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj. Border Complexity of Symbolic Determinant Under Rank One Restriction. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.2,
author = {Chatterjee, Abhranil and Ghosh, Sumanta and Gurjar, Rohit and Raj, Roshan},
title = {{Border Complexity of Symbolic Determinant Under Rank One Restriction}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {2:1--2:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.2},
URN = {urn:nbn:de:0030-drops-182721},
doi = {10.4230/LIPIcs.CCC.2023.2},
annote = {Keywords: Border Complexity, Symbolic Determinant, Valuated Matroid}
}
Document
On Correlation Bounds Against Polynomials
Abstract
We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over 𝔽₂. Our main contributions include:
1) In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible.
2) We propose a new approach for proving correlation bounds with the central "mod functions," consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results."
3) We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions (x_1,… ,x_n) → z^{∑ x_i} for any z on the complex unit circle, and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction.
4) We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric.
Cite as
Peter Ivanov, Liam Pavlovic, and Emanuele Viola. On Correlation Bounds Against Polynomials. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 3:1-3:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{ivanov_et_al:LIPIcs.CCC.2023.3,
author = {Ivanov, Peter and Pavlovic, Liam and Viola, Emanuele},
title = {{On Correlation Bounds Against Polynomials}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {3:1--3:35},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.3},
URN = {urn:nbn:de:0030-drops-182734},
doi = {10.4230/LIPIcs.CCC.2023.3},
annote = {Keywords: Correlation bounds, Polynomials}
}
Document
On the Algebraic Proof Complexity of Tensor Isomorphism
Abstract
The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA '17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank.
We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I → BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI.
Cite as
Nicola Galesi, Joshua A. Grochow, Toniann Pitassi, and Adrian She. On the Algebraic Proof Complexity of Tensor Isomorphism. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 4:1-4:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{galesi_et_al:LIPIcs.CCC.2023.4,
author = {Galesi, Nicola and Grochow, Joshua A. and Pitassi, Toniann and She, Adrian},
title = {{On the Algebraic Proof Complexity of Tensor Isomorphism}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {4:1--4:40},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.4},
URN = {urn:nbn:de:0030-drops-182748},
doi = {10.4230/LIPIcs.CCC.2023.4},
annote = {Keywords: Algebraic proof complexity, Tensor Isomorphism, Graph Isomorphism, Polynomial Calculus, Sum-of-Squares, reductions, lower bounds, proof complexity of linear algebra}
}
Document
Generative Models of Huge Objects
Abstract
This work initiates the systematic study of explicit distributions that are indistinguishable from a single exponential-size combinatorial object. In this we extend the work of Goldreich, Goldwasser and Nussboim (SICOMP 2010) that focused on the implementation of huge objects that are indistinguishable from the uniform distribution, satisfying some global properties (which they coined truthfulness). Indistinguishability from a single object is motivated by the study of generative models in learning theory and regularity lemmas in graph theory. Problems that are well understood in the setting of pseudorandomness present significant challenges and at times are impossible when considering generative models of huge objects.
We demonstrate the versatility of this study by providing a learning algorithm for huge indistinguishable objects in several natural settings including: dense functions and graphs with a truthfulness requirement on the number of ones in the function or edges in the graphs, and a version of the weak regularity lemma for sparse graphs that satisfy some global properties. These and other results generalize basic pseudorandom objects as well as notions introduced in algorithmic fairness. The results rely on notions and techniques from a variety of areas including learning theory, complexity theory, cryptography, and game theory.
Cite as
Lunjia Hu, Inbal Rachel Livni Navon, and Omer Reingold. Generative Models of Huge Objects. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{hu_et_al:LIPIcs.CCC.2023.5,
author = {Hu, Lunjia and Livni Navon, Inbal Rachel and Reingold, Omer},
title = {{Generative Models of Huge Objects}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {5:1--5:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.5},
URN = {urn:nbn:de:0030-drops-182758},
doi = {10.4230/LIPIcs.CCC.2023.5},
annote = {Keywords: pseudorandomness, generative models, regularity lemma}
}
Document
Bounded Relativization
Abstract
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ℭ, we say that a statement is ℭ-relativizing if the statement holds relative to every oracle 𝒪 ∈ ℭ. It is easy to see that every result that relativizes also ℭ-relativizes for every complexity class ℭ. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing.
First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0, BPE^{MCSP}/2^{εn} ⊈ SIZE[2ⁿ/n].
We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021).
Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ≠ L. For example:
- Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ≠ BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ≠ L.
- Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ≠ L.
- Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ≠ L.
In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ≠ EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible.
Cite as
Shuichi Hirahara, Zhenjian Lu, and Hanlin Ren. Bounded Relativization. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 6:1-6:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{hirahara_et_al:LIPIcs.CCC.2023.6,
author = {Hirahara, Shuichi and Lu, Zhenjian and Ren, Hanlin},
title = {{Bounded Relativization}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {6:1--6:45},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.6},
URN = {urn:nbn:de:0030-drops-182764},
doi = {10.4230/LIPIcs.CCC.2023.6},
annote = {Keywords: relativization, circuit lower bound, derandomization, explicit construction, pseudodeterministic algorithms, interactive proofs}
}
Document
Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields
Abstract
For every prime p > 0, every n > 0 and κ = O(log n), we show the existence of an unsatisfiable system of polynomial equations over O(n log n) variables of degree O(log n) such that any Polynomial Calculus refutation over 𝔽_p with M extension variables, each depending on at most κ original variables requires size exp(Ω(n²)/10^κ(M + n log n))
Cite as
Russell Impagliazzo, Sasank Mouli, and Toniann Pitassi. Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{impagliazzo_et_al:LIPIcs.CCC.2023.7,
author = {Impagliazzo, Russell and Mouli, Sasank and Pitassi, Toniann},
title = {{Lower Bounds for Polynomial Calculus with Extension Variables over Finite Fields}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {7:1--7:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.7},
URN = {urn:nbn:de:0030-drops-182774},
doi = {10.4230/LIPIcs.CCC.2023.7},
annote = {Keywords: Proof complexity, Algebraic proof systems, Polynomial Calculus, Extension variables, AC⁰\lbrackp\rbrack-Frege}
}
Document
Spectral Expanding Expanders
Abstract
Dinitz, Schapira, and Valadarsky [Dinitz et al., 2017] introduced the intriguing notion of expanding expanders - a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [Dinitz et al., 2017] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [Bilu and Linial, 2006]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [Dinitz et al., 2017] is to construct spectral expanding expanders (SEE).
In this work, we resolve this question by constructing SEE with spectral expansion which, like [Bilu and Linial, 2006], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [Dinitz et al., 2017], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts.
Cite as
Gil Cohen and Itay Cohen. Spectral Expanding Expanders. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{cohen_et_al:LIPIcs.CCC.2023.8,
author = {Cohen, Gil and Cohen, Itay},
title = {{Spectral Expanding Expanders}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {8:1--8:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.8},
URN = {urn:nbn:de:0030-drops-182780},
doi = {10.4230/LIPIcs.CCC.2023.8},
annote = {Keywords: Expanders, Normalized Random Walk, Spectral Analysis}
}
Document
Hardness Against Linear Branching Programs and More
Abstract
In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is a branching program where each node is allowed to make 𝔽₂-linear queries, and is read-once in the sense that the queries on each path is linearly independent. As their main result, they constructed an explicit function with average-case complexity 2^{n/3-o(n)} against a slightly restricted model, which they call strongly read-once linear branching programs. The main tool in their lower bound result is a new type of extractor, called directional affine extractors, that they introduced.
Our main result is an explicit function with 2^{n-o(n)} average-case complexity against the strongly read-once linear branching program model, which is almost optimal. This result is based on a new connection from this problem to sumset extractors, which is a randomness extractor model introduced by Chattopadhyay and Li (STOC '16) as a generalization of many other well-studied models including two-source extractors, affine extractors and small-space extractors. With this new connection, our lower bound naturally follows from a recent construction of sumset extractors by Chattopadhyay and Liao (STOC '22). In addition, we show that directional affine extractors imply sumset extractors in a restricted setting. We observe that such restricted sumset sources are enough to derive lower bounds, and obtain an arguably more modular proof of the lower bound by Gryaznov, Pudlák and Talebanfard.
We also initiate a study of pseudorandomness against linear branching programs. Our main result here is a hitting set generator construction against regular linear branching programs with constant width. We derive this result based on a connection to Kakeya sets over finite fields.
Cite as
Eshan Chattopadhyay and Jyun-Jie Liao. Hardness Against Linear Branching Programs and More. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 9:1-9:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{chattopadhyay_et_al:LIPIcs.CCC.2023.9,
author = {Chattopadhyay, Eshan and Liao, Jyun-Jie},
title = {{Hardness Against Linear Branching Programs and More}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {9:1--9:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.9},
URN = {urn:nbn:de:0030-drops-182794},
doi = {10.4230/LIPIcs.CCC.2023.9},
annote = {Keywords: linear branching programs, circuit lower bound, sumset extractors, hitting sets}
}
Document
An Improved Trickle down Theorem for Partite Complexes
Abstract
We prove a strengthening of the trickle down theorem for partite complexes. Given a (d+1)-partite d-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are (1-δ)/d-(one-sided) spectral expanders, then the link of any face of co-dimension k is an O((1-δ)/(kδ))-(one-sided) spectral expander, for all 3 ≤ k ≤ d+1. For an application, using our theorem as a black-box, we show that links of faces of co-dimension k in recent constructions of bounded degree high dimensional expanders have spectral expansion at most O(1/k) fraction of the spectral expansion of the links of the worst faces of co-dimension 2.
Cite as
Dorna Abdolazimi and Shayan Oveis Gharan. An Improved Trickle down Theorem for Partite Complexes. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{abdolazimi_et_al:LIPIcs.CCC.2023.10,
author = {Abdolazimi, Dorna and Oveis Gharan, Shayan},
title = {{An Improved Trickle down Theorem for Partite Complexes}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {10:1--10:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.10},
URN = {urn:nbn:de:0030-drops-182807},
doi = {10.4230/LIPIcs.CCC.2023.10},
annote = {Keywords: Simplicial complexes, High dimensional expanders, Trickle down theorem, Bounded degree high dimensional expanders, Locally testable codes, Random walks}
}
Document
Derandomization with Minimal Memory Footprint
Abstract
Existing proofs that deduce BPL = 𝐋 from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE[S] ⊆ DSPACE[c ⋅ S] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1.
Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC⁰, that were not known before.
Cite as
Dean Doron and Roei Tell. Derandomization with Minimal Memory Footprint. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{doron_et_al:LIPIcs.CCC.2023.11,
author = {Doron, Dean and Tell, Roei},
title = {{Derandomization with Minimal Memory Footprint}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {11:1--11:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.11},
URN = {urn:nbn:de:0030-drops-182816},
doi = {10.4230/LIPIcs.CCC.2023.11},
annote = {Keywords: derandomization, space-bounded computation, catalytic space}
}
Document
Improved Learning from Kolmogorov Complexity
Abstract
Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in P/poly, under the uniform distribution and with membership queries. It is still an open problem to get from natural properties learning algorithms that do not rely on membership queries but rather use randomly drawn labeled examples.
Natural properties may be understood as an average-case version of MCSP, the problem of deciding the minimum size of a circuit computing a given truth-table. Problems related to MCSP include those concerning time-bounded Kolmogorov complexity. MKTP, for example, asks for the KT-complexity of a given string. KT-complexity is a relaxation of circuit size, as it does away with the requirement that a short description of a string be interpreted as a boolean circuit. In this work, under assumptions of MKTP and the related problem MK^tP being easy on average, we get learning algorithms for boolean functions in P/poly that
- work over any distribution D samplable by a family of polynomial-size circuits (given explicitly in the case of MKTP),
- only use randomly drawn labeled examples from D, and
- are agnostic (do not require the target function to belong to the hypothesis class). Our results build upon the recent work of Hirahara and Nanashima (FOCS, 2021) who showed similar learning consequences but under a stronger assumption that NP is easy on average.
Cite as
Halley Goldberg and Valentine Kabanets. Improved Learning from Kolmogorov Complexity. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 12:1-12:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{goldberg_et_al:LIPIcs.CCC.2023.12,
author = {Goldberg, Halley and Kabanets, Valentine},
title = {{Improved Learning from Kolmogorov Complexity}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {12:1--12:29},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.12},
URN = {urn:nbn:de:0030-drops-182825},
doi = {10.4230/LIPIcs.CCC.2023.12},
annote = {Keywords: learning, Kolmogorov complexity, meta-complexity, average-case complexity}
}
Document
New Lower Bounds Against Homogeneous Non-Commutative Circuits
Abstract
We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree d which requires homogeneous non-commutative circuit of size Ω(d/log d). For an n-variate polynomial with n > 1, the result can be improved to Ω(nd), if d ≤ n, or Ω(nd (log n)/(log d)), if d ≥ n. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.
Cite as
Prerona Chatterjee and Pavel Hrubeš. New Lower Bounds Against Homogeneous Non-Commutative Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 13:1-13:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.13,
author = {Chatterjee, Prerona and Hrube\v{s}, Pavel},
title = {{New Lower Bounds Against Homogeneous Non-Commutative Circuits}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {13:1--13:10},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.13},
URN = {urn:nbn:de:0030-drops-182835},
doi = {10.4230/LIPIcs.CCC.2023.13},
annote = {Keywords: Algebraic circuit complexity, Non-Commutative Circuits, Homogeneous Computation, Lower bounds against algebraic circuits}
}
Document
On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors
Abstract
Locally Decodable Codes (LDCs) are error-correcting codes C:Σⁿ → Σ^m, encoding messages in Σⁿ to codewords in Σ^m, with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length m that is super-polynomial in n, for codes with constant query complexity and constant alphabet size.
In a very surprising result, Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (SICOMP 2006) show how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs.
In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Standard LDCs for Insdel errors were first studied by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), and are further motivated by recent advances in DNA random access bio-technologies.
Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries (even adaptively), over the binary alphabet. This answers a question explicitly raised by Gur and Lachish (SICOMP 2021) and is the first exponential lower bound for RLDCs. Combined with the results of Ben-Sasson et al., our result exhibits a "phase-transition"-type behavior on the codeword length for some constant-query complexity. We achieve these lower bounds via a transformation of RLDCs to standard Hamming LDCs, using a careful analysis of restrictions of message bits that fix codeword bits.
We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with almost linear codeword length and constant query complexity, matching the parameters of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.
Cite as
Alexander R. Block, Jeremiah Blocki, Kuan Cheng, Elena Grigorescu, Xin Li, Yu Zheng, and Minshen Zhu. On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 14:1-14:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{block_et_al:LIPIcs.CCC.2023.14,
author = {Block, Alexander R. and Blocki, Jeremiah and Cheng, Kuan and Grigorescu, Elena and Li, Xin and Zheng, Yu and Zhu, Minshen},
title = {{On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {14:1--14:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.14},
URN = {urn:nbn:de:0030-drops-182847},
doi = {10.4230/LIPIcs.CCC.2023.14},
annote = {Keywords: Relaxed Locally Decodable Codes, Hamming Errors, Insdel Errors, Lower Bounds}
}
Document
Near-Optimal Set-Multilinear Formula Lower Bounds
Abstract
The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas.
In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP) defined over Θ(n²) variables and of degree Θ(n), such that any product-depth Δ set-multilinear formula computing f has size at least n^Ω(n^{1/Δ}/Δ). Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n^{Ω(log n)}.
If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022).
Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM 2009).
By known depth-reduction results, our lower bounds are essentially tight for f and in general, for any hard polynomial that is in set-multilinear VBP or set-multilinear VP. Any asymptotic improvement in the lower bound (for a hard polynomial, say, in VNP) would imply super-polynomial lower bounds for general set-multilinear circuits.
Cite as
Deepanshu Kush and Shubhangi Saraf. Near-Optimal Set-Multilinear Formula Lower Bounds. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 15:1-15:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{kush_et_al:LIPIcs.CCC.2023.15,
author = {Kush, Deepanshu and Saraf, Shubhangi},
title = {{Near-Optimal Set-Multilinear Formula Lower Bounds}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {15:1--15:33},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.15},
URN = {urn:nbn:de:0030-drops-182855},
doi = {10.4230/LIPIcs.CCC.2023.15},
annote = {Keywords: Algebraic Complexity, Set-multilinear, Formula Lower Bounds}
}
Document
Matrix Multiplication and Number on the Forehead Communication
Abstract
Three-player Number On the Forehead communication may be thought of as a three-player Number In the Hand promise model, in which each player is given the inputs that are supposedly on the other two players' heads, and promised that they are consistent with the inputs of the other players. The set of all allowed inputs under this promise may be thought of as an order-3 tensor. We surprisingly observe that this tensor is exactly the matrix multiplication tensor, which is widely studied in the design of fast matrix multiplication algorithms.
Using this connection, we prove a number of results about both Number On the Forehead communication and matrix multiplication, each by using known results or techniques about the other. For example, we show how the Laser method, a key technique used to design the best matrix multiplication algorithms, can also be used to design communication protocols for a variety of problems. We also show how known lower bounds for Number On the Forehead communication can be used to bound properties of the matrix multiplication tensor such as its zeroing out subrank. Finally, we substantially generalize known methods based on slice-rank for studying communication, and show how they directly relate to the matrix multiplication exponent ω.
Cite as
Josh Alman and Jarosław Błasiok. Matrix Multiplication and Number on the Forehead Communication. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{alman_et_al:LIPIcs.CCC.2023.16,
author = {Alman, Josh and B{\l}asiok, Jaros{\l}aw},
title = {{Matrix Multiplication and Number on the Forehead Communication}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {16:1--16:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.16},
URN = {urn:nbn:de:0030-drops-182861},
doi = {10.4230/LIPIcs.CCC.2023.16},
annote = {Keywords: Number on the forehead, communication complexity, matrix multiplication}
}
Document
Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols
Abstract
A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits.
Recently, Chen and Tell (FOCS'21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance.
In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n^a. We show that if every Arthur-Merlin protocol that runs in time n^c for c = O(log² a) can only compute f correctly on finitely many inputs, then AM is in NP. Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations.
As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM, an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM, as well as an unconditional mild derandomization result for AM.
Cite as
Dieter van Melkebeek and Nicollas Mocelin Sdroievski. Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 17:1-17:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{vanmelkebeek_et_al:LIPIcs.CCC.2023.17,
author = {van Melkebeek, Dieter and Mocelin Sdroievski, Nicollas},
title = {{Instance-Wise Hardness Versus Randomness Tradeoffs for Arthur-Merlin Protocols}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {17:1--17:36},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.17},
URN = {urn:nbn:de:0030-drops-182870},
doi = {10.4230/LIPIcs.CCC.2023.17},
annote = {Keywords: Hardness versus randomness tradeoff, Arthur-Merlin protocol, targeted hitting set generator}
}
Document
Tight Correlation Bounds for Circuits Between AC0 and TC0
Abstract
We initiate the study of generalized AC⁰ circuits comprised of arbitrary unbounded fan-in gates which only need to be constant over inputs of Hamming weight ≥ k (up to negations of the input bits), which we denote GC⁰(k). The gate set of this class includes biased LTFs like the k-OR (outputs 1 iff ≥ k bits are 1) and k-AND (outputs 0 iff ≥ k bits are 0), and thus can be seen as an interpolation between AC⁰ and TC⁰.
We establish a tight multi-switching lemma for GC⁰(k) circuits, which bounds the probability that several depth-2 GC⁰(k) circuits do not simultaneously simplify under a random restriction. We also establish a new depth reduction lemma such that coupled with our multi-switching lemma, we can show many results obtained from the multi-switching lemma for depth-d size-s AC⁰ circuits lifts to depth-d size-s^{.99} GC⁰(.01 log s) circuits with no loss in parameters (other than hidden constants).
Our result has the following applications:
- Size-2^Ω(n^{1/d}) depth-d GC⁰(Ω(n^{1/d})) circuits do not correlate with parity (extending a result of Håstad (SICOMP, 2014)).
- Size-n^Ω(log n) GC⁰(Ω(log² n)) circuits with n^{.249} arbitrary threshold gates or n^{.499} arbitrary symmetric gates exhibit exponentially small correlation against an explicit function (extending a result of Tan and Servedio (RANDOM, 2019)).
- There is a seed length O((log m)^{d-1}log(m/ε)log log(m)) pseudorandom generator against size-m depth-d GC⁰(log m) circuits, matching the AC⁰ lower bound of Håstad up to a log log m factor (extending a result of Lyu (CCC, 2022)).
- Size-m GC⁰(log m) circuits have exponentially small Fourier tails (extending a result of Tal (CCC, 2017)).
Cite as
Vinayak M. Kumar. Tight Correlation Bounds for Circuits Between AC0 and TC0. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 18:1-18:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{kumar:LIPIcs.CCC.2023.18,
author = {Kumar, Vinayak M.},
title = {{Tight Correlation Bounds for Circuits Between AC0 and TC0}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {18:1--18:40},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.18},
URN = {urn:nbn:de:0030-drops-182885},
doi = {10.4230/LIPIcs.CCC.2023.18},
annote = {Keywords: AC⁰, TC⁰, Switching Lemma, Lower Bounds, Correlation Bounds, Circuit Complexity}
}
Document
Criticality of AC⁰-Formulae
Abstract
Rossman [In Proc. 34th Comput. Complexity Conf., 2019] introduced the notion of criticality. The criticality of a Boolean function f : {0,1}ⁿ → {0,1} is the minimum λ ≥ 1 such that for all positive integers t and all p ∈ [0,1], Pr_{ρ∼ℛ_p}[DT_{depth}(f|_ρ) ≥ t] ≤ (pλ)^t, where ℛ_p refers to the distribution of p-random restrictions.
Håstad’s celebrated switching lemma shows that the criticality of any k-DNF is at most O(k). Subsequent improvements to correlation bounds of AC⁰-circuits against parity showed that the criticality of any AC⁰-circuit of size S and depth d+1 is at most O(log S)^d and any regular AC⁰-formula of size S and depth d+1 is at most O((1/d)⋅log S)^d. We strengthen these results by showing that the criticality of any AC⁰-formula (not necessarily regular) of size S and depth d+1 is at most O((log S)/d)^d, resolving a conjecture due to Rossman.
This result also implies Rossman’s optimal lower bound on the size of any depth-d AC⁰-formula computing parity [Comput. Complexity, 27(2):209-223, 2018.]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved #SAT algorithm for AC⁰-formulae.
Cite as
Prahladh Harsha, Tulasimohan Molli, and Ashutosh Shankar. Criticality of AC⁰-Formulae. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{harsha_et_al:LIPIcs.CCC.2023.19,
author = {Harsha, Prahladh and Molli, Tulasimohan and Shankar, Ashutosh},
title = {{Criticality of AC⁰-Formulae}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {19:1--19:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.19},
URN = {urn:nbn:de:0030-drops-182898},
doi = {10.4230/LIPIcs.CCC.2023.19},
annote = {Keywords: AC⁰ circuits, AC⁰ formulae, criticality, switching lemma, correlation bounds}
}
Document
Radical Sylvester-Gallai Theorem for Tuples of Quadratics
Abstract
We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020].
Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022].
Cite as
Abhibhav Garg, Rafael Oliveira, Shir Peleg, and Akash Kumar Sengupta. Radical Sylvester-Gallai Theorem for Tuples of Quadratics. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 20:1-20:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{garg_et_al:LIPIcs.CCC.2023.20,
author = {Garg, Abhibhav and Oliveira, Rafael and Peleg, Shir and Sengupta, Akash Kumar},
title = {{Radical Sylvester-Gallai Theorem for Tuples of Quadratics}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {20:1--20:30},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.20},
URN = {urn:nbn:de:0030-drops-182903},
doi = {10.4230/LIPIcs.CCC.2023.20},
annote = {Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Document
Reducing Tarski to Unique Tarski (In the Black-Box Model)
Abstract
We study the problem of finding a Tarski fixed point over the k-dimensional grid [n]^k. We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point.
Cite as
Xi Chen, Yuhao Li, and Mihalis Yannakakis. Reducing Tarski to Unique Tarski (In the Black-Box Model). In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{chen_et_al:LIPIcs.CCC.2023.21,
author = {Chen, Xi and Li, Yuhao and Yannakakis, Mihalis},
title = {{Reducing Tarski to Unique Tarski (In the Black-Box Model)}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {21:1--21:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.21},
URN = {urn:nbn:de:0030-drops-182919},
doi = {10.4230/LIPIcs.CCC.2023.21},
annote = {Keywords: Tarski fixed point, Query complexity, TFNP}
}
Document
A Distribution Testing Oracle Separating QMA and QCMA
Abstract
It is a long-standing open question in quantum complexity theory whether the definition of non-deterministic quantum computation requires quantum witnesses (QMA) or if classical witnesses suffice (QCMA). We make progress on this question by constructing a randomized classical oracle separating the respective computational complexity classes. Previous separations [Aaronson and Kuperberg, 2007; Bill Fefferman and Shelby Kimmel, 2018] required a quantum unitary oracle. The separating problem is deciding whether a distribution supported on regular un-directed graphs either consists of multiple connected components (yes instances) or consists of one expanding connected component (no instances) where the graph is given in an adjacency-list format by the oracle. Therefore, the oracle is a distribution over n-bit boolean functions.
Cite as
Anand Natarajan and Chinmay Nirkhe. A Distribution Testing Oracle Separating QMA and QCMA. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 22:1-22:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{natarajan_et_al:LIPIcs.CCC.2023.22,
author = {Natarajan, Anand and Nirkhe, Chinmay},
title = {{A Distribution Testing Oracle Separating QMA and QCMA}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {22:1--22:27},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.22},
URN = {urn:nbn:de:0030-drops-182928},
doi = {10.4230/LIPIcs.CCC.2023.22},
annote = {Keywords: quantum non-determinism, complexity theory}
}
Document
Translationally Invariant Constraint Optimization Problems
Abstract
We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the computational complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class FP^NEXP. Gottesman and Irani (FOCS 2009) also studied classical constraint satisfaction problems using this strong notion of translational-invariance; they show that the problem of deciding whether the cost of the optimal assignment is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel’s famous result from 1988, showing that the function version of SAT is complete for the class FP^NP.
An essential ingredient in the proof is a study of the computational complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of Ω(N^(1/4)), where the grid size is N × N. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. This might be of independent interest. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for P^NEXP.
Cite as
Dorit Aharonov and Sandy Irani. Translationally Invariant Constraint Optimization Problems. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{aharonov_et_al:LIPIcs.CCC.2023.23,
author = {Aharonov, Dorit and Irani, Sandy},
title = {{Translationally Invariant Constraint Optimization Problems}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {23:1--23:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.23},
URN = {urn:nbn:de:0030-drops-182932},
doi = {10.4230/LIPIcs.CCC.2023.23},
annote = {Keywords: Constraint satisfaction, Tiling, Translational-invariance}
}
Document
An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree
Abstract
While it is known that there is at most a polynomial separation between quantum query complexity and the polynomial degree for total functions, the precise relationship between the two is not clear for partial functions.
In this paper, we demonstrate an exponential separation between exact polynomial degree and approximate quantum query complexity for a partial Boolean function. For an unbounded alphabet size, we have a constant versus polynomial separation.
Cite as
Andris Ambainis and Aleksandrs Belovs. An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{ambainis_et_al:LIPIcs.CCC.2023.24,
author = {Ambainis, Andris and Belovs, Aleksandrs},
title = {{An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {24:1--24:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.24},
URN = {urn:nbn:de:0030-drops-182943},
doi = {10.4230/LIPIcs.CCC.2023.24},
annote = {Keywords: Polynomials, Quantum Adversary Bound, Separations in Query Complexity}
}
Document
Trade-Offs Between Entanglement and Communication
Abstract
We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on n bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every k ≥ ~1:
Q‖^* versus R2^*: We show that quantum simultaneous protocols with Θ̃(k⁵log³n) qubits of entanglement can exponentially outperform two-way randomized protocols with O(k) qubits of entanglement. This resolves an open problem from [Dmitry Gavinsky, 2008] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gavinsky, 2019; Girish et al., 2022].
R‖^* versus Q‖^*: We show that classical simultaneous protocols with Θ̃(k log n) qubits of entanglement can exponentially outperform quantum simultaneous protocols with O(k) qubits of entanglement, resolving an open question from [Gavinsky et al., 2006; Gavinsky, 2019]. The best result prior to our work was a relational separation against protocols without entanglement [Gavinsky et al., 2006].
R‖^* versus R1^*: We show that classical simultaneous protocols with Θ̃(k log n) qubits of entanglement can exponentially outperform randomized one-way protocols with O(k) qubits of entanglement. Prior to our work, only a relational separation was known [Dmitry Gavinsky, 2008].
Cite as
Srinivasan Arunachalam and Uma Girish. Trade-Offs Between Entanglement and Communication. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 25:1-25:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{arunachalam_et_al:LIPIcs.CCC.2023.25,
author = {Arunachalam, Srinivasan and Girish, Uma},
title = {{Trade-Offs Between Entanglement and Communication}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {25:1--25:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.25},
URN = {urn:nbn:de:0030-drops-182957},
doi = {10.4230/LIPIcs.CCC.2023.25},
annote = {Keywords: quantum, communication complexity, exponential separation, boolean hidden matching, forrelation, xor lemma}
}
Document
New Sampling Lower Bounds via the Separator
Abstract
Suppose that a target distribution can be approximately sampled by a low-depth decision tree, or more generally by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the sampler so that its output distribution is still not too far from the target distribution, and at the same time many output coordinates are almost pairwise independent.
This new tool is then used to obtain several new sampling lower bounds and separations, including a separation between AC0 and low-depth decision trees, and a hierarchy theorem for sampling. It is also used to obtain a new proof of the Patrascu-Viola data-structure lower bound for Rank, thereby unifying sampling and data-structure lower bounds.
Cite as
Emanuele Viola. New Sampling Lower Bounds via the Separator. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{viola:LIPIcs.CCC.2023.26,
author = {Viola, Emanuele},
title = {{New Sampling Lower Bounds via the Separator}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {26:1--26:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.26},
URN = {urn:nbn:de:0030-drops-182967},
doi = {10.4230/LIPIcs.CCC.2023.26},
annote = {Keywords: Sampling, data structures, lower bounds, cell probe, decision forest, AC0, rank, predecessor}
}
Document
A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs
Abstract
We define a novel notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it.
We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and n^{k/2} / ε² constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+O_k(ε) fraction of constraints.
Previously, this was known for even k, but for odd k one needed n^{k/2} (log n)^{O(1)} / ε² random constraints to achieve the same conclusion.
Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck’s inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n^⌈k/2⌉) and the third one cannot work when the number of constraints is o(n^{k/2} √{log n}).
We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with n^{k/2} / ε² constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.
Cite as
Tommaso d'Orsi and Luca Trevisan. A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{dorsi_et_al:LIPIcs.CCC.2023.27,
author = {d'Orsi, Tommaso and Trevisan, Luca},
title = {{A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {27:1--27:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.27},
URN = {urn:nbn:de:0030-drops-182979},
doi = {10.4230/LIPIcs.CCC.2023.27},
annote = {Keywords: CSP, k-XOR, strong refutation, sum-of-squares, tensor, graph, hypergraph, non-backtracking walk}
}
Document
Towards Optimal Depth-Reductions for Algebraic Formulas
Abstract
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula.
Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas.
This improves on previous results in the regime when d is small (i.e., d = s^o(1)). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting.
We also show that these results cannot be improved in the monotone setting, even for commutative formulas.
Cite as
Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, and Sébastien Tavenas. Towards Optimal Depth-Reductions for Algebraic Formulas. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{fournier_et_al:LIPIcs.CCC.2023.28,
author = {Fournier, Herv\'{e} and Limaye, Nutan and Malod, Guillaume and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
title = {{Towards Optimal Depth-Reductions for Algebraic Formulas}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {28:1--28:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.28},
URN = {urn:nbn:de:0030-drops-182986},
doi = {10.4230/LIPIcs.CCC.2023.28},
annote = {Keywords: Algebraic formulas, depth-reduction}
}
Document
Constant-Depth Circuits vs. Monotone Circuits
Abstract
We establish new separations between the power of monotone and general (non-monotone) Boolean circuits:
- For every k ≥ 1, there is a monotone function in AC⁰ (constant-depth poly-size circuits) that requires monotone circuits of depth Ω(log^k n). This significantly extends a classical result of Okol'nishnikova [Okol'nishnikova, 1982] and Ajtai and Gurevich [Ajtai and Gurevich, 1987]. In addition, our separation holds for a monotone graph property, which was unknown even in the context of AC⁰ versus mAC⁰.
- For every k ≥ 1, there is a monotone function in AC⁰[⊕] (constant-depth poly-size circuits extended with parity gates) that requires monotone circuits of size exp(Ω(log^k n)). This makes progress towards a question posed by Grigni and Sipser [Grigni and Sipser, 1992]. These results show that constant-depth circuits can be more efficient than monotone formulas and monotone circuits when computing monotone functions.
In the opposite direction, we observe that non-trivial simulations are possible in the absence of parity gates: every monotone function computed by an AC⁰ circuit of size s and depth d can be computed by a monotone circuit of size 2^{n - n/O(log s)^{d-1}}. We show that the existence of significantly faster monotone simulations would lead to breakthrough circuit lower bounds. In particular, if every monotone function in AC⁰ admits a polynomial size monotone circuit, then NC² is not contained in NC¹.
Finally, we revisit our separation result against monotone circuit size and investigate the limits of our approach, which is based on a monotone lower bound for constraint satisfaction problems (CSPs) established by Göös, Kamath, Robere and Sokolov [Göös et al., 2019] via lifting techniques. Adapting results of Schaefer [Thomas J. Schaefer, 1978] and Allender, Bauland, Immerman, Schnoor and Vollmer [Eric Allender et al., 2009], we obtain an unconditional classification of the monotone circuit complexity of Boolean-valued CSPs via their polymorphisms. This result and the consequences we derive from it might be of independent interest.
Cite as
Bruno P. Cavalar and Igor C. Oliveira. Constant-Depth Circuits vs. Monotone Circuits. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 29:1-29:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{cavalar_et_al:LIPIcs.CCC.2023.29,
author = {Cavalar, Bruno P. and Oliveira, Igor C.},
title = {{Constant-Depth Circuits vs. Monotone Circuits}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {29:1--29:37},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.29},
URN = {urn:nbn:de:0030-drops-182998},
doi = {10.4230/LIPIcs.CCC.2023.29},
annote = {Keywords: circuit complexity, monotone circuit complexity, bounded-depth circuis, constraint-satisfaction problems}
}
Document
A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph
Abstract
We prove that the degree 4 sum-of-squares (SOS) relaxation of the clique number of the Paley graph on a prime number p of vertices has value at least Ω(p^{1/3}). This is in contrast to the widely believed conjecture that the actual clique number of the Paley graph is O(polylog(p)). Our result may be viewed as a derandomization of that of Deshpande and Montanari (2015), who showed the same lower bound (up to polylog(p) terms) with high probability for the Erdős-Rényi random graph on p vertices, whose clique number is with high probability O(log(p)). We also show that our lower bound is optimal for the Feige-Krauthgamer construction of pseudomoments, derandomizing an argument of Kelner. Finally, we present numerical experiments indicating that the value of the degree 4 SOS relaxation of the Paley graph may scale as O(p^{1/2 - ε}) for some ε > 0, and give a matrix norm calculation indicating that the pseudocalibration construction for SOS lower bounds for random graphs will not immediately transfer to the Paley graph. Taken together, our results suggest that degree 4 SOS may break the "√p barrier" for upper bounds on the clique number of Paley graphs, but prove that it can at best improve the exponent from 1/2 to 1/3.
Cite as
Dmitriy Kunisky and Xifan Yu. A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 30:1-30:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{kunisky_et_al:LIPIcs.CCC.2023.30,
author = {Kunisky, Dmitriy and Yu, Xifan},
title = {{A Degree 4 Sum-Of-Squares Lower Bound for the Clique Number of the Paley Graph}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {30:1--30:25},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.30},
URN = {urn:nbn:de:0030-drops-183008},
doi = {10.4230/LIPIcs.CCC.2023.30},
annote = {Keywords: convex optimization, sum of squares, Paley graph, derandomization}
}
Document
Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem
Abstract
We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}ⁿ → {0,1}, SoS requires degree Ω(s^{1-ε}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ⊈ P/poly.
We also show that for any 0 < α < 1 there are Boolean functions with circuit complexity larger than 2^{n^α} but SoS requires size 2^{2^Ω(n^α)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions.
Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q.
Cite as
Per Austrin and Kilian Risse. Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{austrin_et_al:LIPIcs.CCC.2023.31,
author = {Austrin, Per and Risse, Kilian},
title = {{Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {31:1--31:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.31},
URN = {urn:nbn:de:0030-drops-183011},
doi = {10.4230/LIPIcs.CCC.2023.31},
annote = {Keywords: Proof Complexity, Sum of Squares, Minimum Circuit Size Problem}
}
Document
Leakage-Resilient Hardness vs Randomness
Abstract
A central open problem in complexity theory concerns the question of whether all efficient randomized algorithms can be simulated by efficient deterministic algorithms. The celebrated "hardness v.s. randomness” paradigm pioneered by Blum-Micali (SIAM JoC’84), Yao (FOCS’84) and Nisan-Wigderson (JCSS’94) presents hardness assumptions under which e.g., prBPP = prP (so-called "high-end derandomization), or prBPP ⊆ prSUBEXP (so-called "low-end derandomization), and more generally, under which prBPP ⊆ prDTIME(𝒞) where 𝒞 is a "nice" class (closed under composition with a polynomial), but these hardness assumptions are not known to also be necessary for such derandomization.
In this work, following the recent work by Chen and Tell (FOCS’21) that considers "almost-all-input" hardness of a function f (i.e., hardness of computing f on more than a finite number of inputs), we consider "almost-all-input" leakage-resilient hardness of a function f - that is, hardness of computing f(x) even given, say, √|x| bits of leakage of f(x). We show that leakage-resilient hardness characterizes derandomization of prBPP (i.e., gives a both necessary and sufficient condition for derandomization), both in the high-end and in the low-end setting.
In more detail, we show that there exists a constant c such that for every function T, the following are equivalent:
- prBPP ⊆ prDTIME(poly(T(poly(n))));
- Existence of a poly(T(poly(n)))-time computable function f :{0,1}ⁿ → {0,1}ⁿ that is almost-all-input leakage-resilient hard with respect to n^c-time probabilistic algorithms. As far as we know, this is the first assumption that characterizes derandomization in both the low-end and the high-end regime.
Additionally, our characterization naturally extends also to derandomization of prMA, and also to average-case derandomization, by appropriately weakening the requirements on the function f. In particular, for the case of average-case (a.k.a. "effective") derandomization, we no longer require the function to be almost-all-input hard, but simply satisfy the more standard notion of average-case leakage-resilient hardness (w.r.t., every samplable distribution), whereas for derandomization of prMA, we instead consider leakage-resilience for relations.
Cite as
Yanyi Liu and Rafael Pass. Leakage-Resilient Hardness vs Randomness. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{liu_et_al:LIPIcs.CCC.2023.32,
author = {Liu, Yanyi and Pass, Rafael},
title = {{Leakage-Resilient Hardness vs Randomness}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {32:1--32:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.32},
URN = {urn:nbn:de:0030-drops-183022},
doi = {10.4230/LIPIcs.CCC.2023.32},
annote = {Keywords: Derandomization, Leakage-Resilient Hardness}
}
Document
On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation
Abstract
Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time T for the simulation greatly affect algorithm runtime as expected. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time o(T), for some large classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time T. On the other hand, while there exist lower bounds of Ω(T) circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. As a result, physical systems with system size scaling with T can potentially do a fast-forwarding simulation. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism.
In this work, we give a negative result for the above open problem in various settings. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth o(T). In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians on n qubits that cannot be simulated via an oracle circuit of depth o(T/n^c), where the Hamiltonians act on n qubits, and c is a constant. Lastly, we generalize the above results and show that any simulators that are geometrically local Hamiltonians cannot do the simulation much faster than parallel quantum algorithms.
Cite as
Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 33:1-33:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{chia_et_al:LIPIcs.CCC.2023.33,
author = {Chia, Nai-Hui and Chung, Kai-Min and Hsieh, Yao-Ching and Lin, Han-Hsuan and Lin, Yao-Ting and Shen, Yu-Ching},
title = {{On the Impossibility of General Parallel Fast-Forwarding of Hamiltonian Simulation}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {33:1--33:45},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.33},
URN = {urn:nbn:de:0030-drops-183038},
doi = {10.4230/LIPIcs.CCC.2023.33},
annote = {Keywords: Hamiltonian simulation, Depth lower bound, Parallel query lower bound}
}
Document
The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate
Abstract
Variational Quantum Algorithms (VQAs), such as the Quantum Approximate Optimization Algorithm (QAOA) of [Farhi, Goldstone, Gutmann, 2014], have seen intense study towards near-term applications on quantum hardware. A crucial parameter for VQAs is the depth of the variational ansatz used - the smaller the depth, the more amenable the ansatz is to near-term quantum hardware in that it gives the circuit a chance to be fully executed before the system decoheres. In this work, we show that approximating the optimal depth for a given VQA ansatz is intractable. Formally, we show that for any constant ε > 0, it is QCMA-hard to approximate the optimal depth of a VQA ansatz within multiplicative factor N^(1-ε), for N denoting the encoding size of the VQA instance. (Here, Quantum Classical Merlin-Arthur (QCMA) is a quantum generalization of NP.) We then show that this hardness persists in the even "simpler" QAOA-type settings. To our knowledge, this yields the first natural QCMA-hard-to-approximate problems.
Cite as
Lennart Bittel, Sevag Gharibian, and Martin Kliesch. The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 34:1-34:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{bittel_et_al:LIPIcs.CCC.2023.34,
author = {Bittel, Lennart and Gharibian, Sevag and Kliesch, Martin},
title = {{The Optimal Depth of Variational Quantum Algorithms Is QCMA-Hard to Approximate}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {34:1--34:24},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.34},
URN = {urn:nbn:de:0030-drops-183045},
doi = {10.4230/LIPIcs.CCC.2023.34},
annote = {Keywords: Variational quantum algorithms (VQA), Quantum Approximate Optimization Algorithm (QAOA), circuit depth minimization, Quantum-Classical Merlin-Arthur (QCMA), hardness of approximation, hybrid quantum algorithms}
}
Document
An Algorithmic Approach to Uniform Lower Bounds
Abstract
We propose a new family of circuit-based sampling tasks, such that non-trivial algorithmic solutions to certain tasks from this family imply frontier uniform lower bounds such as "NP is not in uniform ACC⁰" and "NP does not have uniform polynomial-size depth-two threshold circuits". Indeed, the most general versions of our sampling tasks have implications for central open problems such as NP vs P and PSPACE vs P.
We argue the soundness of our approach by showing that the non-trivial algorithmic solutions we require do follow from standard cryptographic assumptions. In addition, we give evidence that a version of our approach for uniform circuits is necessary in order to separate NP from P or PSPACE from P. We give an algorithmic characterization for the PSPACE vs P question: PSPACE ≠ P iff either E has sub-exponential time non-uniform algorithms infinitely often or there are non-trivial space-efficient solutions to our sampling tasks for uniform Boolean circuits.
We show how to use our framework to capture uniform versions of known non-uniform lower bounds, as well as classical uniform lower bounds such as the space hierarchy theorem and Allender’s uniform lower bound for the Permanent. We also apply our framework to prove new lower bounds: NP does not have polynomial-size uniform AC⁰ circuits with a bottom layer of MOD 6 gates, nor does it have polynomial-size uniform AC⁰ circuits with a bottom layer of threshold gates.
Our proofs exploit recently defined probabilistic time-bounded variants of Kolmogorov complexity [Zhenjian Lu et al., 2022; Halley Goldberg et al., 2022; Halley Goldberg et al., 2022].
Cite as
Rahul Santhanam. An Algorithmic Approach to Uniform Lower Bounds. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 35:1-35:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{santhanam:LIPIcs.CCC.2023.35,
author = {Santhanam, Rahul},
title = {{An Algorithmic Approach to Uniform Lower Bounds}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {35:1--35:26},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.35},
URN = {urn:nbn:de:0030-drops-183053},
doi = {10.4230/LIPIcs.CCC.2023.35},
annote = {Keywords: Probabilistic Kolmogorov complexity, sampling algorithms, uniform lower bounds}
}
Document
Colourful TFNP and Propositional Proofs
Abstract
Recent work has shown that many of the standard TFNP classes - such as PLS, PPADS, PPAD, SOPL, and EOPL - have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C-PLS, C-PPADS, C-PPAD, C-SOPL, and C-EOPL. While C-PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that:
- Each of the classes C-PLS, C-PPADS, and C-SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C-PLS is characterized by depth-1.5 Frege (Res(polylog(n)).
- The classes C-PPAD and C-EOPL coincide exactly with the uncoloured classes PPADS and SOPL, respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively.
- Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C-PLS ∩ C-PPADS = C-SOPL, generalizing the intersection theorem PLS ∩ PPADS = SOPL. However, while it is known in the uncoloured world that PLS ∩ PPAD = EOPL = CLS, we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C-PLS^O ∩ C-PPAD^O ⊋ C-EOPL^O. To prove our results, we introduce an abstract multivalued proof system - the Blockwise Calculus - which may be of independent interest.
Cite as
Ben Davis and Robert Robere. Colourful TFNP and Propositional Proofs. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 36:1-36:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
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@InProceedings{davis_et_al:LIPIcs.CCC.2023.36,
author = {Davis, Ben and Robere, Robert},
title = {{Colourful TFNP and Propositional Proofs}},
booktitle = {38th Computational Complexity Conference (CCC 2023)},
pages = {36:1--36:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-282-2},
ISSN = {1868-8969},
year = {2023},
volume = {264},
editor = {Ta-Shma, Amnon},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.36},
URN = {urn:nbn:de:0030-drops-183066},
doi = {10.4230/LIPIcs.CCC.2023.36},
annote = {Keywords: oracle separations, TFNP, proof complexity, Res(k), lower bounds}
}