The study of the complexity of counting problems has a long and rich history in theoretical computer science. Counting problems (and closely related sampling problems) arise naturally in many different fields, for example in statistical physics they correspond to partition functions and for studies of the equilibrium states of idealized models of physical systems, and in Bayesian inference they arise for the study of posterior distributions or maximum likelihood distributions. The specific questions addressed here are long-standing open problems, progress on which will be of wide interest. The project will develop new tools for approximate counting and is likely to make new and useful connections between statistical physics, probability and computational complexity. The research results will be disseminated via course notes, a summer school and workshops. Any practical algorithms that result will be made publicly available.
The overall goal of the project is to extend the known boundary of polynomial-time tractability for counting problems, to understand whether randomness is essential and how it could be eliminated, and to push the limits of the current fastest randomized algorithms towards practicality. Specific aims include: (1) Polynomial-time randomized approximation schemes for some fundamental problems that have thus far eluded efficient solutions, (2) Deterministic polynomial-time approximation schemes for some central problems that have celebrated randomized algorithms and (3) Faster randomized algorithms for classical counting problems.
