Counting Perfect Matchings in Dense Graphs Is Hard

10/26/2022

∙

by   Nicolas El Maalouly, et al.

∙

0

∙

share

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 ≤ 2, or general graphs of independence number ≤ 2. Our proof is by reduction from counting perfect matchings in bipartite graphs, via elementary linear algebra tricks and graph constructions.