# Matching is as Easy as the Decision Problem, in the NC Model

## TL;DR

This paper establishes that solving the matching problem in NC reduces to solving its decision problem, and provides an NC algorithm for minimum-weight perfect matching given an oracle for the decision problem, marking a significant advance in parallel algorithms.

## Contribution

It introduces an NC algorithm for minimum-weight perfect matching assuming an oracle for the decision problem, advancing the understanding of matching in parallel computation.

## Key findings

- An NC algorithm for minimum-weight perfect matching with an oracle for the decision problem.
- Reduction of the main open problem to the decision problem in NC.
- Extension of pseudo-deterministic RNC algorithms from bipartite to general graphs.

## Abstract

Is matching in NC, i.e., is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in TCS for over three decades, ever since the discovery of randomized NC matching algorithms [KUW85, MVV87]. Over the last five years, the theoretical computer science community has launched a relentless attack on this question, leading to the discovery of several powerful ideas. We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is given an oracle for the decision problem. Consequently, for settling the main open problem, it suffices to obtain an NC algorithm for the decision problem. We believe this new fact has qualitatively changed the nature of this open problem. All known efficient matching algorithms for general graphs follow one of two approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based algorithm follows a new approach and uses many of the ideas discovered in the last five years. The difficulty of obtaining an NC perfect matching algorithm led researchers to study matching vis-a-vis clever relaxations of the class NC. In this vein, recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC algorithm with the additional requirement that on the same graph, it should return the same (i.e., unique) perfect matching for almost all choices of random bits. A corollary of our reduction is an analogous algorithm for general graphs.