Congested Clique Counting for Local Gibbs Distributions

Abstract

There are well established reductions between combinatorial sampling and counting problems (Jerrum, Valiant, Vazirani TCS 1986). Building off of a very recent parallel algorithm utilizing this connection (Liu, Yin, Zhang arxiv 2024), we demonstrate the first approximate counting algorithm in the CongestedClique for a wide range of problems. Most interestingly, we present an algorithm for approximating the number of $q$-colorings of a graph within $\epsilon$-multiplicative error, when $q>\alpha\Delta$ for any constant $\alpha>2$, in $\Tilde{O}\big(\frac{n^{1/3}}{\epsilon^2}\big)$ rounds. More generally, we achieve a runtime of $\Tilde{O}\big(\frac{n^{1/3}}{\epsilon^2}\big)$ rounds for approximating the partition function of Gibbs distributions defined over graphs when simple locality and fast mixing conditions hold. Gibbs distributions are widely used in fields such as machine learning and statistical physics. We obtain our result by providing an algorithm to draw $n$ random samples from a distributed Markov chain in parallel, using similar ideas to triangle counting (Dolev, Lenzen, Peled DISC 2012) and semiring matrix multiplication (Censor-Hillel, Kaski, Korhonen, Lenzen, Paz, Suomela PODC 2015). Aside from counting problems, this result may be interesting for other applications requiring a large number of samples. In the special case of estimating the partition function of the hardcore model, also known as counting weighted independent sets, we can do even better and achieve an $\Tilde{O}\big(\frac{1}{\epsilon^2}\big)$ round algorithm, when the fugacity $\lambda \leq \frac{\alpha}{\Delta-1}$, where $\alpha$ is an arbitrary constant less than $1$.