Exact Matching and the Top-k Perfect Matching Problem

TL;DR

This paper demonstrates the polynomial-time equivalence between the longstanding Exact Matching problem and the Top-k Perfect Matching problem, connecting two important problems in combinatorial optimization.

Contribution

It provides a reduction showing that solving one problem efficiently implies an efficient solution for the other, establishing their polynomial-time equivalence.

Findings

Exact Matching reduces to Top-k Perfect Matching in polynomial time

The two problems are polynomial-time equivalent

No known deterministic polynomial algorithm for Exact Matching

Abstract

The aim of this note is to provide a reduction of the Exact Matching problem to the Top-$k$ Perfect Matching Problem. Together with earlier work by El Maalouly, this shows that the two problems are polynomial-time equivalent. The Exact Matching Problem is a well-known 40 years old problem for which a randomized, but no deterministic poly-time algorithm has been discovered. The Top-$k$ Perfect Matching Problem is the problem of finding a perfect matching which maximizes the total weight of the $k$ heaviest edges contained in it.