The Cost of Perfection for Matchings in Graphs

TL;DR

This paper investigates the ratio between maximum weight perfect matchings and general matchings in graphs, providing bounds and characterizations, with applications in computer graphics and specific graph classes.

Contribution

It characterizes graphs with extreme ratios, establishes lower bounds for bridgeless cubic graphs, and provides tight bounds for subclasses and bipartite graphs.

Findings

Characterization of graphs with extreme matching ratios

Lower bounds for bridgeless cubic graphs

Tight bounds for subclasses and bipartite graphs

Abstract

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs.