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)

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Dagstuhl Service Team

Documents

Dagstuhl Report, Volume 7, Issue 8 Dagstuhl Report
Aims & Scope
List of Participants
Shared Documents
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.)
License
  Creative Commons BY 3.0 Unported license
  Ivona Bezáková, Leslie Ann Goldberg, and Mark R. Jerrum

Dagstuhl Seminar Series

Classification

  • Data Structures / Algorithms / Complexity

Keywords

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

Book exhibition

Books from the participants of the current Seminar 

Book exhibition in the library, ground floor, during the seminar week.

Documentation

In the series Dagstuhl Reports each Dagstuhl Seminar and Dagstuhl Perspectives Workshop is documented. The seminar organizers, in cooperation with the collector, prepare a report that includes contributions from the participants' talks together with a summary of the seminar.

 

Download overview leaflet (PDF).

Publications

Furthermore, a comprehensive peer-reviewed collection of research papers can be published in the series Dagstuhl Follow-Ups.

Dagstuhl's Impact

Please inform us when a publication was published as a result from your seminar. These publications are listed in the category Dagstuhl's Impact and are presented on a special shelf on the ground floor of the library.

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