Counting Perfect Matchings in Dense Graphs Is Hard

TL;DR

This paper proves that counting perfect matchings remains computationally hard (#P-complete) even in very dense graphs, resolving a conjecture and extending the understanding of problem complexity in dense graph classes.

Contribution

It demonstrates the #P-completeness of counting perfect matchings in dense graphs, including bipartite graphs with bipartite independence number ≤ 2, using novel reductions and elementary linear algebra techniques.

Findings

Counting perfect matchings is #P-complete in dense bipartite graphs.

The problem remains hard in graphs with independence number ≤ 2.

The proof involves reductions from bipartite graph matchings and linear algebra tricks.

Abstract

We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number $\leq 2$, or general graphs of independence number $\leq 2$. Our proof is by reduction from counting perfect matchings in bipartite graphs, via elementary linear algebra tricks and graph constructions.