Parameterized Complexity and Approximation Issues for the Colorful Components Problems

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 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.

Finding a Small Number of Colourful Components

“…Hence, the problems Colourful Partition and Colourful Components are equivalent for trees. The following hardness and FPT results are due to Bruckner et al [3] and Dondi and Sikora [7]. Note that trees of diameter at most 3 are stars and double stars (the graph obtained from two stars by adding an edge between their central vertices), for which both problems are readily seen to be polynomial-time solvable.…”

“…A subdivided star is the graph obtained by subdividing the edges of a star. In fact, Dondi and Sikora [7] found a cubic kernel for coloured trees when parameterized by k and also gave an O * (1.554 k )-time exact algorithm for coloured trees. In addition to Theorem 6, Bruckner et al [3] showed that Colourful Components is FPT for general coloured graphs when parameterized by the number of colours ℓ and the number of edge deletions p.…”

“…For Statement (7), note that each bag X i of size 1, and, by Claim 1, every bag X j such that j is a descendent of i is always respected, so again all values of x w for w ∈ V [i] are identical. Finally, Statements (8) and (9) follow from Claim 5.…”

“…When Zheng et al [17] considered this model, one approach they followed was to try to delete as few edges as possible such that the connected components of the resulting graph G ′ are colourful, that is, contain no more than one vertex of any colour; in this case the connected components give the partition of V (G). This led them to the Orthogonal Partition problem, introduced in [8] and also known as the Colourful Components problem [1,3,7]. (Note that a graph is colourful if each of its components is colourful.…”

Parameterized complexity and approximation issues for the colorful components problems

“…Finally, we show in Section 7 that results similar to those of Section 4, hold also for MEC. A preliminary version of this work appeared in [9].…”