Approximation and FPT Algorithms for Finding DM-Irreducible Spanning Subgraphs

TL;DR

A generalization related to the Dulmage--Mendelsohn decompositions of bipartite graphs instead of the strong connectivity of directed graphs is considered, and these approximation and FPT results to the generalized setting are extended.

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 Jájá (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.

Figures (1)

• Figure 1: A tight example of our $2$-approximation algorithm for DMISS.

Theorems & Definitions (29)

• Lemma 2.1: cf. frank2011connections
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5