How to Secure Matchings Against Edge Failures

TL;DR

This paper addresses the problem of securing bipartite matchings against edge failures by developing algorithms for adding minimum cost edges, including approximation, fixed parameter, and polynomial-time algorithms for specific graph classes.

Contribution

It introduces new algorithms for edge augmentation to protect matchings, including approximation and fixed parameter algorithms, and characterizes graph classes with polynomial solutions.

Findings

Logarithmic-factor approximation algorithm for unit-cost case

Polynomial-time algorithm for chordal-bipartite graphs

Fixed parameter algorithm based on treewidth

Abstract

Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We provide efficient exact and approximation algorithms. In particular, for the unit-cost problem, we provide a $\log_2 n$-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we settle the approximability of the problem and show a close relation to the Directed Steiner Forest Problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. Our methods rely on a close relationship to the classical strong connectivity augmentation problem and directed Steiner problems.