Maximum cuts in edge-colored graphs

Abstract

The input of the Maximum Colored Cut problem consists of a graph G = (V, E) with an edge-coloring c : E → {1, 2, 3, . . . , p} and a positive integer k, and the question is whether G has a nontrivial edge cut using at least k colors. The Colorful Cut problem has the same input but asks for a nontrivial edge cut using all p colors. Unlike what happens for the classical Maximum Cut problem, we prove that both problems are NP-complete even on complete, planar, or bounded treewidth graphs. Furthermore, we prove that Colorful Cut is NP-complete even when each color class induces a clique of size at most 3, but is trivially solvable when each color induces a K 2 . On the positive side, we prove that Maximum Colored Cut is fixedparameter tractable when parameterized by either k or p, by constructing a cubic kernel in both cases.

The edge-labeled survivable network design problem: Formulations and branch-and-cut

“…The presented algorithms are able to find all the optimal solutions described in the literature. In [20], Faria et al study the maximum labeled cut problem. Given an edge-labeled graph and a parameter k, the question is whether or not G has a nontrivial edge cut using at least k labels (colors).…”

Maximum Colored Cuts in Edge‐Colored Complete Graphs

“…maximizing the number of colors of edges going across V 1 and V 2 . Faria et al [3] proposed this problem and studied its complexity. Let m |E(G)|.…”