Approximation and FPT Algorithms for Finding DM-Irreducible Spanning Subgraphs

TL;DR

This paper extends approximation and fixed-parameter tractability results from the classic strongly connected spanning subgraph problem to a generalized setting involving Dulmage--Mendelsohn decompositions of bipartite graphs, offering new algorithms and insights.

Contribution

It generalizes existing approximation and FPT algorithms for directed graphs to bipartite graph decompositions, broadening the scope of these methods.

Findings

Extended 2-approximation algorithms to the generalized setting.

Proved fixed-parameter tractability for the generalized problem.

Connected the problem to Dulmage--Mendelsohn decompositions.

Abstract

Finding a minimum-weight strongly connected spanning subgraph of an edge-weighted directed graph is equivalent to the weighted version of the well-known strong connectivity augmentation problem. This problem is NP-hard, and a simple $2$-approximation algorithm was proposed by Frederickson and J\'aj\'a (1981); surprisingly, it still achieves the best known approximation ratio in general. Also, Bang-Jensen and Yeo (2008) showed that the unweighted problem is FPT (fixed-parameter tractable) parameterized by the difference from a trivial upper bound of the optimal value. In this paper, we consider a generalization related to the Dulmage--Mendelsohn decompositions of bipartite graphs instead of the strong connectivity of directed graphs, and extend these approximation and FPT results to the generalized setting.