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Counting Perfect Matchings in Dense Graphs Is Hard (2210.15014v1)
Published 26 Oct 2022 in cs.DS and math.CO
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.