Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs

TL;DR

This paper presents a deterministic method to assign weights ensuring unique perfect matchings in certain bipartite graphs, extending previous results and placing the problem in the SPL complexity class.

Contribution

It derandomizes the Isolation Lemma for $K_{3,3}$-free and $K_5$-free bipartite graphs, previously known only for planar bipartite graphs.

Findings

Perfect matching problem for these graphs is in SPL.

Provides a deterministic log-space construction of weights.

Offers an alternative proof for reachability in these graph classes.

Abstract

The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph, ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for $K_{3,3}$-free and $K_5$-free bipartite graphs, i.e. we give a deterministic log-space construction of such a weight assignment for these graphs. Such a construction was known previously for planar bipartite graphs. Our result implies that the perfect matching problem for $K_{3,3}$-free and $K_5$-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result -- reachability for $K_{3,3}$-free and $K_5$-free graphs is in UL.