Exploiting Independent Subformulas: A Faster Approximation Scheme for #k-SAT

TL;DR

This paper introduces a faster randomized approximation scheme for #k-SAT that leverages independent subformulas to significantly reduce search space, improving efficiency for counting solutions in Boolean k-CNF formulas.

Contribution

The paper presents a novel method to identify independent substructures in #k-SAT formulas, enabling more efficient approximation algorithms across all k values.

Findings

For #3-SAT, runtime is O(ε^{-2}*1.51426^n).

For #4-SAT, runtime is O(ε^{-2}*1.60816^n).

The approach reduces search space by exploiting independent subformulas.

Abstract

We present an improvement on Thurley's recent randomized approximation scheme for #k-SAT where the task is to count the number of satisfying truth assignments of a Boolean function {\Phi} given as an n-variable k-CNF. We introduce a novel way to identify independent substructures of {\Phi} and can therefore reduce the size of the search space considerably. Our randomized algorithm works for any k. For #3-SAT, it runs in time O(\epsilon^{-2}*1.51426^n), for #4-SAT, it runs in time O(\epsilon^{-2}*1.60816^n), with error bound \epsilon.