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The Cartographer
Counting Perfect Matchings in Dense Graphs Is Hard
October 26, 2022 Β· Declared Dead Β· π arXiv.org
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Authors
Nicolas El Maalouly, Yanheng Wang
arXiv ID
2210.15014
Category
cs.DS: Data Structures & Algorithms
Cross-listed
math.CO
Citations
2
Venue
arXiv.org
Last Checked
4 months ago
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.
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