### 20.08.17 - 25.08.17, Seminar 17341

# Computational Counting

### The following text appeared on our web pages prior to the seminar, and was included as part of the invitation.

## Motivation

Computational counting problems arise in practical applications in many fields such as statistical physics, machine learning and computational biology. 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 complexity of computational counting problems requires a coherent set of techniques which is different in flavour from techniques found 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 searching for efficient (possibly randomised) approximation algorithms, or parameterised algorithms that are efficient when some key parameter is "small". Often, the range of applicability of approximation algorithms is limited by the existence of "phase transitions" as some parameter varies. Great progress has been made in recent years towards understanding the complexity of approximate counting, based largely on this connection with phase transitions.

Specific themes of this Dagstuhl Seminar include, but are not limited to, the following.

- 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.
- Counting constraint satisfaction problems and holants. The partition functions of many models in statistical physics are included within this setting.

**License**

Creative Commons BY 3.0 Unported license

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