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 $c:E\to {1,2,3,\ldots , p}$ 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 fixed-parameter tractable when parameterized by either $k$ or $p$, by constructing a cubic kernel in both cases.