Parameterized Local Search for Max $c$-Cut

In the NP-hard Max $c$-Cut problem, one is given an undirected edge-weighted graph $G$ and aims to color the vertices of $G$ with $c$ colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with $c=2$ is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS Max $c$-Cut where we are also given a vertex coloring and an integer $k$ and the task is to find a better coloring that changes the color of at most $k$ vertices, if such a coloring exists; otherwise, the given coloring is $k$-optimal. We show that, for all $c\ge 2$, LS Max $c$-Cut presumably cannot be solved in $f(k)\cdot n^{\mathcal{O}(1)}$ time even on bipartite graphs. We then present an algorithm for LS Max $c$-Cut with running time $\mathcal{O}((3e\Delta)^k\cdot c\cdot k^3\cdot\Delta\cdot n)$, where $\Delta$ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for a state-of-the-art heuristic for Max $c$-Cut. We show that using parameterized local search, the results of this state-of-the-art heuristic can be further improved on a set of standard benchmark instances.