Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

TL;DR

This paper introduces a deterministic method for assigning weights to bipartite planar graphs to uniquely identify perfect matchings, leading to improved parallel algorithms for matching problems.

Contribution

It provides the first deterministic weight assignment for bipartite planar graphs that guarantees a unique minimum perfect matching, improving parallel complexity bounds.

Findings

Achieves deterministic isolation of perfect matchings in bipartite planar graphs.

Reduces matching decision and construction problems to matrix singularity testing.

Provides a highly parallel SPL algorithm with improved bounds for bipartite planar graphs.

Abstract

We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by (Allender et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple.