https://www.dagstuhl.de/17341

### August 20 – 25 , 2017, Dagstuhl Seminar 17341

# Computational Counting

## Organizers

Ivona Bezáková (Rochester Institute of Technology, US)

Leslie Ann Goldberg (University of Oxford, GB)

Mark R. Jerrum (Queen Mary University of London, GB)

## For support, please contact

## Documents

Dagstuhl Report, Volume 7, Issue 8

Aims & Scope

List of Participants

Dagstuhl's Impact: Documents available

Dagstuhl Seminar Schedule [pdf]

## Summary

Computational counting problems arise in practical applications in many fields such as statistical physics, information theory and machine learning. In such a problem, the goal is to compute or to estimate a weighted sum. Some typical computational counting problems include evaluating a probability, the expectation of a random variable, a partition function, or an integral.

The study of the computational complexity of counting problems requires a coherent set of techniques which are different in flavour from those employed in other algorithmic branches of computer science. Relevant techniques include the analysis of Markov chains, the analysis of correlation decay, parameterised algorithms and complexity, and dichotomy techniques for constructing detailed classifications.

Most computational problems are intractable when considered from the perspective of classical complexity, so it is important to find ways to cope with intractability. These include approximation, randomisation, as well as viewing computational counting through the lens of parameterised complexity, where the goal is to find algorithms that are efficient when some key parameter is "small". Intractability thresholds often relate to "phase transitions" as these key parameters vary. Great progress has been made in recent years towards understanding the complexity of approximate counting, based largely on a connection with these phase transitions.

Specific themes identified for consideration at the meeting included:

*Exact counting*, including classifications, quasi-polynomial and/or moderately exponential algorithms for intractable problems, and parameterised algorithms; also complexity-theoretic limitations to obtaining exact solutions.*Approximate counting*including Markov Chain Monte Carlo (*MCMC*) algorithms, and algorithms based on decay of correlations; also complexity-theoretic limitations to obtaining approximate solutions.- The interplay between
*phase transitions*and computational tractability. *Constraint satisfaction*problems and the more general*Holant*framework. The partition functions of many models in statistical physics are included within this setting.

In the event, the talks ranged more widely than this list suggests.

Although the topic of Computational Counting has been explored at various meetings, including at Dagstuhl, for a number of years, it continues to retain its freshness. New approaches are found, new insights are gained, and new connections drawn with other areas both inside and outside computer science. Among the new directions that have emerged since the previous Dagstuhl Seminar in this series are the following.

- Results from quantum information theory applied to the apparently unrelated task of classifying the complexity of Holant problems. (Refer to the talk by Miriam Backens.)
- A new paradigm for designing polynomial-time algorithms for approximating partition functions with complex parameters. This is based on Taylor expansion in a zero-free region of the parameter space combined with an ingenious approach to enumerating small substructures. (Refer to talks by Alexander Barvinok, Jingcheng Liu, Viresh Patel and Guus Regts.)
- Emerging connections between the Lovász Local Lemma - specifically the Shearer condition and the Moser-Tardos algorithmic version - and sampling and approximate counting. (Refer to talks by Andreas Galanis and Heng Guo.)

**Summary text license**

Creative Commons BY 3.0 Unported license

Ivona Bezáková, Leslie Ann Goldberg, and Mark R. Jerrum

## Dagstuhl Seminar Series

- 21151: "Counting and Sampling: Algorithms and Complexity" (2021)
- 13031: "Computational Counting" (2013)
- 10481: "Computational Counting" (2010)

## Classification

- Data Structures / Algorithms / Complexity

## Keywords

- Approximation algorithms
- Computational complexity
- Counting problems
- Partition functions
- Phase transitions