Maximium Priority Matchings

TL;DR

This paper introduces an efficient algorithm to find maximum priority matchings in graphs, optimizing the allocation based on vertex priorities using a modified augmenting path approach.

Contribution

It presents a novel variation of Edmonds' algorithm that computes maximum priority matchings in O(mn) time, improving upon existing methods.

Findings

Algorithm achieves maximum priority matching efficiently.

Time complexity is O(mn).

Applicable to graphs with vertex priorities.

Abstract

Let $G=(V,E)$ be an undirected graph with $n$ vertices and $m$ edges, in which each vertex $u$ is assigned an integer priority in $[1,n]$, with 1 being the "highest" priority. Let $M$ be a matching of $G$. We define the priority score of $M$ to be an $n$-ary integer in which the $i$-th most-significant digit is the number of vertices with priority $i$ that are incident to an edge in $M$. We describe a variation of the augmenting path method (Edmonds' algorithm) that finds a matching with maximum priority score in $O(mn)$ time.