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 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 − ε, 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.

Towards a Complexity Dichotomy for Colourful Components Problems on k-caterpillars and Small-Degree Planar Graphs

“…Nonetheless, we propose a linear time algorithm for Colourful Components and Colourful Partition on 1-caterpillars and cyclic 1-caterpillars with unbounded degree in Section 2.2. This result improves the complexity of the known quadratic-time algorithm for paths [6] and widens the class of graphs. We, therefore, obtain a complete complexity dichotomy for the problems on k-caterpillars with regard to k and the maximum degree in the graph.…”

“…However, this is not the case on general graphs [5]. Both problems are known to be NP-complete on subdivided stars [6], trees of diameter at most 4 [4], and trees with maximum degree 6 [5]. Trees of diameter at most 4 are in fact a subclass of 2-caterpillars, so both problems are NP-complete on 2-caterpillars when the maximum degree is unbounded.…”

“…Minimising the number of similar genes in different subsets of the partition is a well studied variant [4,5,8,13,15], and it corresponds to minimising the number of edges between the colourful components (as in Colourful Components). Alternatively, one can ask for a partition of minimum size, i.e which contains the minimum number of orthologous genes, or equivalently the minimum number of colourful components [1,5,6] (as in Colourful Partition). Another variant, not studied in this paper, considers the objective function that maximises the number of edges in the transitive closure [1,6,13].…”

“…Alternatively, one can ask for a partition of minimum size, i.e which contains the minimum number of orthologous genes, or equivalently the minimum number of colourful components [1,5,6] (as in Colourful Partition). Another variant, not studied in this paper, considers the objective function that maximises the number of edges in the transitive closure [1,6,13]. Now, we give the formal definitions of the problems considered herein.…”