On the Complexity of Weighted Greedy Matchings

TL;DR

This paper proves that computing maximum weight greedy matchings is strongly NP-hard and APX-complete, introduces a randomized approximation algorithm for special graph classes, and explores connections to maximum cardinality matching approximations.

Contribution

It establishes the computational hardness of GreedyMatching, introduces a randomized approximation algorithm for bush graphs, and links its performance to maximum cardinality matching approximations.

Findings

GreedyMatching is strongly NP-hard and APX-complete.

A randomized approximation algorithm (RGMA) is effective on bush graphs.

The approximation ratio of RGMA relates to the approximation of maximum cardinality matching.

Abstract

Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GreedyMatching. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GreedyMatching is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GreedyMatching is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider two natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GreedyMatching on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is $\rho$, then for every $\epsilon>0$ the randomized MRG algorithm of [Aronson et al., Rand. Struct. Alg. 1995] gives a $(\rho-\epsilon)$-approximation for the maximum cardinality matching. We conjecture that a tight bound for $\rho$ is $\frac{2}{3}$; we prove our conjecture true for two subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for $\rho$) is a long-standing open problem [Poloczek and Szegedy; FOCS 2012]. This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem.