Planar Graph Perfect Matching is in NC

TL;DR

This paper presents the first known deterministic parallel NC algorithm for finding perfect matchings in planar graphs, resolving a long-standing open problem in theoretical computer science.

Contribution

It introduces an NC algorithm for perfect matching in planar graphs, leveraging counting techniques and polytope conditions, a breakthrough in parallel algorithms.

Findings

NC algorithm for perfect matching in planar graphs

Uses polytope face conditions to find matchings

Achieves deterministic parallel complexity for this problem

Abstract

Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which $\Omega(n)$ new conditions, involving constraints of the polytope, are simultaneously satisfied. Several other ideas are also needed, such as finding a point in the interior of the minimum weight face of this polytope and finding a balanced tight odd set in NC.