Counting Maximum Matchings in Planar Graphs Is Hard
Computational Complexity
2021-03-09 v2 Combinatorics
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.
Cite
@article{arxiv.2001.01493,
title = {Counting Maximum Matchings in Planar Graphs Is Hard},
author = {Istvan Miklos and Miklos Kresz},
journal= {arXiv preprint arXiv:2001.01493},
year = {2021}
}
Comments
The presented result is not new. A proof of the main theorem appeared in Vadhan (2001) THE COMPLEXITY OF COUNTING IN SPARSE, REGULAR, AND PLANAR GRAPHS