Parameterized Complexity and Approximation Issues for the Colorful Components Problems

TL;DR

This paper investigates the computational complexity and approximation limits of colorful components problems in vertex-colored graphs, providing new algorithms and hardness results for various graph classes and parameters.

Contribution

It introduces fixed-parameter algorithms, kernelization results, and hardness proofs for MCC and MEC problems, expanding understanding of their complexity in different graph settings.

Findings

MCC on trees is equivalent to Minimum Cut on Trees and not approximable within 1.36 - ε.

MCC is polynomial on paths but NP-hard on graphs close to disjoint paths.

Fixed-parameter algorithms and polynomial kernels are developed for MEC and MCC in specific cases.

Abstract

The quest for colorful components (connected components where each color is associated with at most one vertex) inside a vertex-colored graph has been widely considered in the last ten years. Here we consider two variants, Minimum Colorful Components (MCC) and Maximum Edges in transitive Closure (MEC), introduced in 2011 in the context of orthology gene identification in bioinformatics. The input of both MCC and MEC is a vertex-colored graph. MCC asks for the removal of a subset of edges, so that the resulting graph is partitioned in the minimum number of colorful connected components; MEC asks for the removal of a subset of edges, so that the resulting graph is partitioned in colorful connected components and the number of edges in the transitive closure of such a graph is maximized. We study the parameterized and approximation complexity of MCC and MEC, for general and restricted instances. For MCC on trees we show that the problem is basically equivalent to Minimum Cut on Trees, thus MCC is not approximable within factor $1.36 - \varepsilon$, it is fixed-parameter tractable and it admits a poly-kernel (when the parameter is the number of colorful components). Moreover, we show that MCC, while it is polynomial time solvable on paths, it is NP-hard even for graphs with constant distance to disjoint paths number. Then we consider the parameterized complexity of MEC when parameterized by the number $k$ of edges in the transitive closure of a solution (the graph obtained by removing edges so that it is partitioned in colorful connected components). We give a fixed-parameter algorithm for MEC paramterized by $k$ and, when the input graph is a tree, we give a poly-kernel.