Weighted Connected Matchings

TL;DR

This paper investigates the computational complexity of the Maximum Weight Connected Matching problem, revealing NP-hardness in many graph classes, polynomial solvability in some special cases, and parameterized algorithms with kernelization limits.

Contribution

It establishes NP-hardness for weighted connected matchings in various graph classes, provides polynomial algorithms for specific graph types, and explores parameterized complexity and kernelization boundaries.

Findings

NP-hardness in bounded diameter bipartite and planar graphs

Polynomial-time algorithms for trees and subcubic graphs

Single exponential time algorithm parameterized by treewidth

Abstract

A matching $M$ is a $\mathscr{P}$-matching if the subgraph induced by the endpoints of the edges of $M$ satisfies property $\mathscr{P}$. As examples, for appropriate choices of $\mathscr{P}$, the problems Induced Matching, Uniquely Restricted Matching, Connected Matching and Disconnected Matching arise. For many of these problems, finding a maximum $\mathscr{P}$-matching is a knowingly NP-Hard problem, with few exceptions, such as connected matchings, which has the same time complexity as usual Maximum Matching problem. The weighted variant of Maximum Matching has been studied for decades, with many applications, including the well-known Assignment problem. Motivated by this fact, in addition to some recent researches in weighted versions of acyclic and induced matchings, we study the Maximum Weight Connected Matching. In this problem, we want to find a matching $M$ such that the endpoint vertices of its edges induce a connected subgraph and the sum of the edge weights of $M$ is maximum. Unlike the unweighted Connected Matching problem, which is in P for general graphs, we show that Maximum Weight Connected Matching is NP-Hard even for bounded diameter bipartite graphs, starlike graphs, planar bipartite, and bounded degree planar graphs, while solvable in linear time for trees and subcubic graphs. When we restrict edge weights to be non negative only, we show that the problem turns to be polynomially solvable for chordal graphs, while it remains NP-Hard for most of the cases when weights can be negative. Our final contributions are on parameterized complexity. On the positive side, we present a single exponential time algorithm when parameterized by treewidth. In terms of kernelization, we show that, even when restricted to binary weights, Weighted Connected Matching does not admit a polynomial kernel when parameterized by vertex cover under standard complexity-theoretical hypotheses.