Finding Disjoint Paths on Edge-Colored Graphs: More Tractability Results

TL;DR

This paper explores the computational complexity of finding disjoint paths in edge-colored graphs, demonstrating persistent hardness even under structural graph constraints and introducing a new variant with additional complexity challenges.

Contribution

It provides new complexity results for MaxCDP and introduces MaxCDDP, analyzing their tractability and hardness on structured graph classes.

Findings

MaxCDP remains hard to approximate in FPT-time.

MaxCDDP is hard even on graphs close to disjoint paths.

Extended some complexity results from MaxCDP to MaxCDDP.

Abstract

The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (called MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (namely the number of vertices to remove to make the graph a collection of disjoint paths and the size of the vertex cover of the graph), which makes sense since graphs in social networks are not random and have structure. The problem was known to be hard to approximate in polynomial time and not fixed-parameter tractable (FPT) for the natural parameter. Here, we show that it is still hard to approximate, even in FPT-time. Finally, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on graphs at distance two from disjoint paths.