Solving Routing Problems via Important Cuts

TL;DR

This paper presents a new approach using important cuts to develop faster fixed-parameter tractable algorithms for various routing problems, broadening applicability beyond specific graph classes.

Contribution

It introduces important cuts to improve FPT algorithms for multiple routing problems, including the general Minimum Vulnerability problem on all undirected graphs.

Findings

FPT algorithms for Mixed Chinese Postman Problem and Minimum Shared Edges.

FPT algorithm for Minimum Vulnerability on all undirected graphs.

Significantly faster algorithms compared to previous methods.

Abstract

We introduce a novel approach of using important cuts which allowed us to design significantly faster fixed-parameter tractable (FPT) algorithms for the following routing problems: the Mixed Chinese Postman Problem parameterized by the number of directed edges (Gutin et al., JCSS 2017), the Minimum Shared Edges problem (MSE) parameterized by the number p of paths between two specified vertices (Fluschnik et al., JCSS 2019), and the Weighted Min Cut Prevention problem (Gruttemeier et al., WG 2021). The Minimum Vulnerability problem (MV) is a generalization of MSE (Assadi et al., Algorithmica 2014). The only known FPT algorithm for MV parameterized by p (the same parameter as for MSE) was for chordal graphs (Aoki et al., JCO 2018). We design an FPT algorithm for MV on all undirected graphs.