The Parameterized Complexity of Some Minimum Label Problems

Abstract

We study the parameterized complexity of several minimum label graph problems, in which we are given an undirected graph whose edges are labeled, and a property Π, and we are asked to find a subset of edges satisfying property Π that uses the minimum number of labels. These problems have a lot of applications in networking. We show that all the problems under consideration are W[2]-hard when parameterized by the number of used labels, and that they remain W[2]-hard even on graphs whose pathwidth is bounded above by a small constant. On the positive side, we prove that most of these problems are FPT when parameterized by the solution size, that is, the size of the sought edge set. For example, we show that computing a maximum matching or an edge dominating set that uses the minimum number of labels, is FPT when parameterized by the solution size. Proving that some of these problems are FPT is nontrivial, and requires interesting and elegant algorithmic methods that we develop in this paper.

Maximum cuts in edge-colored graphs

“…Complexity issues related to Minimum Colored (s, t)-Cut and Minimum Colored Cut have been widely investigated in recent years (cf. [1,4,10,11,17,27,29,30,31,32]).…”

“…Zhang and Fu [31] showed that Minimum Colored (s, t)-Cut is NP-hard even if the maximum length of any path is equal to two; and when restricted to disjoint-path graphs, Minimum Colored (s, t)-Cut can be solved in polynomial time if the number of edges of each color is at most two. Regarding the parameterized complexity of Minimum Colored (s, t)-Cut, Fellows et al [17] showed that the problem is W[2]-hard when parameterized by the number of colors of the solution, and W[1]-hard when parameterized by the number of edges of the solution. Coudert et al [11] showed that Minimum Colored Cut can be solved in time 2 k · n O(1) , where k is the number of colors with span larger than one.…”

“…• when Π is the property of being a perfect matching of the input graph G, the Maximum (Minimum) Perfect Matching problems have been studied in [17,22];…”

“…• when Π is the property of being a edge dominating set of the input graph G, the Minimum Colored Edge Domination Set problem has been studied in [17];…”

A 2-Approximation for the Minimum Duplication Speciation Problem

“…The Minimum Edge-Cut problem is a special case of the SFM Problem: given a graph H ¼ (V, E), if we denote by g the cut-set function defined from 2 V to R that associates to a subset X of V the number of edges in the edge-cut E H (X), then g is a submodular function. The problem of finding a subset of vertices X & V that minimizes g(X) is then equivalent to the Minimum Edge-Cut problem on H. However, the MLEC Problem is not an SFM Problem, and, due to its relationship with the Minimum Label-Cut problem (Chauve and Ouangraoua, 2009), is in fact NP-complete, even when each edge label appears at most two times in the graph (Zhang et al, 2011;Fellows, 2010).…”