Refined Parameterizations for Computing Colored Cuts in Edge-Colored Graphs

Abstract

In the NP-hard Colored (s,t)-Cut problem, the input is a graph G = (V,E) together with an edge-coloring ℓ : E → C, two vertices s and t, and a number k. The question is whether there is a set $S\subseteq C$ S ⊆ C of at most k colors such that deleting every edge with a color from S destroys all paths between s and t in G. We continue the study of the parameterized complexity of Colored (s,t)-Cut. First, we consider parameters related to the structure of G. For example, we study parameterization by the number ξi of edge deletions that are needed to transform G into a graph with maximum degree i. We show that Colored (s,t)-Cut is W[2]-hard when parameterized by ξ3, but fixed-parameter tractable when parameterized by ξ2. Second, we consider parameters related to the coloring ℓ. We show fixed-parameter tractability for three parameters that are potentially smaller than the total number of colors |C| and provide a linear-size problem kernel for a parameter related to the number of edges with rare edge colors.

Parameterized Algorithms for Colored Clustering

“…It may be useful to consider structural parameters that explicitly take the structure of the coloring into account. Such parameters were studied by Morawietz et al (2022). Unfortunately, most of those parameters are smaller than the number of colors and, therefore, of little use in the context of COLORED CLUSTERING.…”