Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs
Chetan Gupta, Raghunath Tewari, Vimal Raj Sharma
TL;DR
This work addresses the problem of deciding whether a bipartite graph of genus $g$ has a perfect matching by constructing in $O(1)$ space a set of isolating weight functions. It leverages a two-stage strategy: (i) create a linear combination weight function to ensure at most one minimum-weight perfect matching within each of the $2^{2g}$ signature classes defined by the polygonal schema, and (ii) apply the Fredman–Komlós–Szemerédi hashing to produce $k=O(n^{c'}+2^{g})$ functions $w_i$ such that, for some $i$, $w_i$ isolates a minimum-weight perfect matching if one exists; matchings in the oriented graph $\vec{G}$ correspond to matchings in $G$. For $g=O(\log n)$ this yields a polynomially bounded isolating family and, via the ARZ99 reduction, places the perfect matching problem in $SPL$. The approach overcomes the need for nonzero weights on all cycles by first isolating within classes and then across classes, and it generalizes prior path-isolating ideas to the perfect-matching setting on high-genus bipartite graphs, with potential implications for related derandomization questions.
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.