Approximately counting maximal independent set is equivalent to #SAT

TL;DR

This paper establishes a fundamental equivalence between approximately counting maximal independent sets and #SAT, revealing their computational complexity connection and implications for related fields.

Contribution

It proves that the #MIS problem is AP-interreducible with #SAT for general graphs, the first such result establishing their computational equivalence.

Findings

#MIS is #P-complete to count exactly.

Approximate counting of #MIS is computationally equivalent to #SAT.

This equivalence impacts understanding of counting problems in complexity theory.

Abstract

A maximal independent set is an independent set that is not a subset of any other independent set. It is also the key problem of mathematics, computer science, and other fields. A counting problem is a type of computational problem that associated with the number of solutions. Besides, counting problems help us better understand several fields such as algorithm analysis, complexity theory, artificial intelligence, etc. The problem of counting maximal independent sets is #P-complete. So it is natural to think about approximate counting for maximal independent sets problem. In this article, we study the complexity of approximately counting maximal independent sets. Specifically, we are the first to prove that the #MIS problem is AP-interreducible with the #SAT of a given general graph.