Approximate Counting in Local Lemma Regimes

TL;DR

This paper develops efficient approximate counting algorithms for problems in local lemma regimes, including intersection probabilities and subspace dimensions, using cluster expansion methods, with applications to CNF formulas and quantum satisfiability.

Contribution

It introduces novel approximation algorithms for intersection probabilities and subspace dimensions in local lemma regimes, extending to quantum and classical problems.

Findings

Polynomial-time approximation schemes for intersection probability and dimension of commuting projectors.

Efficient algorithms for counting satisfying assignments of CNF formulas.

Approximate dimension computation for quantum satisfiability subspaces.

Abstract

We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is based on the cluster expansion method. We obtain fully polynomial-time approximation schemes for both the probability of intersection and the dimension of intersection for commuting projectors. For general projectors, we provide two algorithms: a fully polynomial-time approximation scheme under a global inclusion-exclusion stability condition, and an efficient affine approximation under a spectral gap assumption. As corollaries of our results, we obtain efficient algorithms for approximating the number of satisfying assignments of conjunctive normal form formulae and the dimension of satisfying subspaces of quantum satisfiability formulae.