Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs

TL;DR

This paper presents a deterministic logspace method to construct a weight function that isolates a perfect matching in bipartite graphs with logarithmic genus, leading to an efficient decision process for perfect matchings in such graphs.

Contribution

It introduces a deterministic logspace construction of an isolating weight function for $O( ext{log } n)$ genus bipartite graphs, enabling perfect matching decision in SPL.

Findings

Decidable whether a perfect matching exists in $O( ext{log } n)$ genus bipartite graphs.

Constructs an isolating weight function in logarithmic space.

Shows that perfect matching decision in these graphs is in SPL.

Abstract

We show that given an embedding of an $O(\log n)$ genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists. As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for $O(\log n)$ genus bipartite graphs.