Towards derandomising Markov chain Monte Carlo

TL;DR

This paper introduces a novel framework for derandomising certain MCMC algorithms, enabling efficient deterministic approximate counting for hypergraph problems by leveraging a new coupling method.

Contribution

The paper proposes a coupling towards the past technique that simplifies derandomisation of MCMC, achieving near state-of-the-art deterministic algorithms for hypergraph counting problems.

Findings

Deterministic algorithms match randomised counterparts in efficiency.

Coupling towards the past evaluates multiple variables in logarithmic time.

Framework applies under local lemma conditions similar to existing methods.

Abstract

We present a new framework to derandomise certain Markov chain Monte Carlo (MCMC) algorithms. As in MCMC, we first reduce counting problems to sampling from a sequence of marginal distributions. For the latter task, we introduce a method called coupling towards the past that can, in logarithmic time, evaluate one or a constant number of variables from a stationary Markov chain state. Since there are at most logarithmic random choices, this leads to very simple derandomisation. We provide two applications of this framework, namely efficient deterministic approximate counting algorithms for hypergraph independent sets and hypergraph colourings, under local lemma type conditions matching, up to lower order factors, their state-of-the-art randomised counterparts.