Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

TL;DR

This paper proves that deciding the existence and uniqueness of perfect matchings in bipartite graphs embedded on surfaces of constant genus can be done in logspace, extending previous results from planar graphs using algebraic topology.

Contribution

It introduces a logspace algorithm for perfect matching problems in bounded genus bipartite graphs, utilizing a novel weight function based on algebraic topology.

Findings

Deciding perfect matching existence is in SPL.

Deciding unique perfect matching is in SPL.

Finding a perfect matching is in FL^SPL.

Abstract

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes $\oplus\L$ (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in $\FL^{\SPL}$. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.