Counting Maximum Matchings in Planar Graphs Is Hard

TL;DR

This paper proves that counting maximum matchings in planar bipartite graphs is #P-complete, revealing computational hardness even in restricted planar cases where perfect matchings are polynomial-time computable.

Contribution

It establishes the #P-completeness of counting maximum matchings in planar bipartite graphs, a previously unknown hardness result for this class.

Findings

Counting maximum matchings in planar bipartite graphs is #P-complete.

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

Counting perfect matchings in planar graphs is polynomial-time solvable.

Abstract

Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.