Maximum Cuts in Edge-colored Graphs

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>0, 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 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 K2. On the positive side, we prove that Maximum Colored Cut is fixed-parameter tractable when parameterized by either k or p, and that it admits a cubic kernel in both cases.