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Dagstuhl Seminar 23111

Computational Complexity of Discrete Problems

( Mar 12 – Mar 17, 2023 )

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Computational complexity studies the amount of resources (such as time, space, randomness, parallelism, or communication) that are necessary to solve computational problems in various models of computation. Finding efficient algorithms for solving computational tasks is crucial in many practical applications. Despite a long line of research, for many discrete problems that arise in practice it is not known if they can be solved efficiently - in particular, in (randomized) polynomial time. While efficient algorithms clearly have obvious applications, knowing that a problem can not be solved efficiently can also have high practical impact. For example, lower bounds on the amount of resources needed to solve specific problems can be used to construct good pseudorandom generators to derandomize probabilistic algorithms. Similarly, the security of our currently used crypto-systems hinges on the assumption that certain discrete problems - like factoring - are hard to solve; and we would very much like to prove that more efficient algorithms cannot exist for factoring. Proving lower bounds is a challenging task since one needs to argue against all possible algorithms. In the last few decades, lower bound methods have been developed for various restricted or special models of computation. These results often involve the use of sophisticated mathematical techniques and despite a lot of effort we still do not have - somewhat frustratingly - strong enough techniques to establish for instance superlinear lower bounds for specific problems in general computational models, such as the model of Boolean circuits. In this Dagstuhl Seminar, which is the most recent incarnation of a seminar series that stretches back many years, we brought together leading experts and talented junior researchers to discuss the most exciting recent developments in different areas of computational complexity of discrete problems - both regarding recent results, but also regarding open problems. In both cases, a particular focus was on lower bounds and on whether and, if so, how ideas and methods from one theory area can yield insights in another theory area.

To enable and encourage discussions between the researchers present in Dagstuhl, time was allotted to three different formats: The presentation and discussion of current research results and methods, the presentation and discussion of open problems and conjectures, and on-site collaborative theory research. Each day of the seminar started with a morning session dedicated to survey talks, talks sharing research results, and talks introducing new techniques. The afternoons were dedicated to collaborative research in various forms: On Tuesday, research was done in break-out sessions in which smaller groups of participants explored different open research questions in depth. The topics covered were the following (in alphabetical order of chairs):

  1. Tree Codes, chaired by Gil Cohen.
  2. Lifting Dichotomies, chaired by Yuval Filmus.
  3. Finding Tarski Fixed-Points, chaired by Kristoffer Hansen.
  4. Hitting Sets Versus Orthogonal Vectors (a.k.a. ksat Versus ∀∃ksat), chaired by Marvin Kühnemann.
  5. Simple Versions of sat, chaired by Till Tantau.

Two other formats were intended to intrigue participants in research questions beyond their own speciality (and succeeded in doing so). Firstly, there were open problem sessions, where Gil Cohen, Yuval Filmus, Mika Göös, Rohit Gurjar, Alexander Kulikov, Jakob Nordström, Rüdiger Reischuk, Robert Robere, and Ben Lee Volk presented different research questions. Secondly, at various points during the seminar, there were short ``talks to talk about'', which introduced exciting recent topics, results, or problems and which gave people intriguing topics to discuss over lunch or dinner. Talks to talk about were given by Sourav Chakraborty, Manaswi Parashar, and Till Tantau.

A final important objective of the seminar was to foster collaborations not only between researchers working on different topics, but also between junior and senior researchers. Towards this aim, talks of more junior researchers were scheduled (as far as possible) on the first day, giving them early exposure and allowing other participants to talk to them about the presented research results during the whole week. Naturally, both junior and senior participants had ample opportunity to socialize, be it during the traditional Wednesday afternoon hike or the wine-and-cheese party on Thursday.

The organizers, Anna Gál, Meena Mahajan, Rahul Santhanam, and Till Tantau, thank all participants for the many contributions they made. We would also like to especially thank the Dagstuhl staff, who were - as usual - extremely friendly, helpful, and professional regarding all organizational matters surrounding the seminar. Finally, we express our great gratitude to Manaswi Paraashar for his invaluable help assembling and preparing the full report.

Copyright Anna Gál, Meena Mahajan, Rahul Santhanam, and Till Tantau


Computational complexity studies the amount of resources (such as time, space, randomness, or communication) necessary to solve computational problems in various models of computation – a crucial task both in theoretical and practical applications. Despite a long line of research, for many practical problems it is not known if they can be solved efficiently. Here, “efficiently” can refer to polynomial-time algorithms, whose existence is not known for problems like Satisfiability or Factoring , but also to cubic or even quadratic time, where it would be important to establish whether these running times are best possible for basic problems, to what extent they can be improved, and whether parallel algorithms allow improvements of the runtime.

These fundamental questions motivate developments in areas from algorithm design to circuit complexity, communication complexity and proof complexity. The seminar’s topics will include new lower bounds on formula size and circuit size, complexity measures of Boolean functions, the algorithmic method for proving lower bounds, fixed-parameter tractability and hardness magnification, communication complexity and lifting techniques, as well as proof complexity. Continuing a long and successful series, this Dagstuhl Seminar will build on previous experience and retain the series’ distinctive features, while also capitalizing on new and exciting developments in the area such as the proof of the Sensitivity Conjecture, new lower bounds and results on matrix rigidity via the algorithmic method, and the remarkable success of lifting techniques in translating lower bounds from query complexity to communication complexity and from proof complexity to circuit complexity.

The bulk of the seminar will be taken up by talks and discussions. The topics will depend on and be driven by the participants, who will share their current research interests in talks, open problem sessions, and smaller group research.

Copyright Anna Gál, Meena Mahajan, Rahul Santhanam, and Till Tantau


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  • Computational Complexity
  • Data Structures and Algorithms

  • computational complexity
  • circuit complexity
  • communication complexity
  • randomness
  • lower bounds