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 wants 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 f and an integer k and the task is to find a better coloring f ′ that differs from f in at most k entries, if such a coloring exists; otherwise, f is k -optimal. We show that, for all c ≥ 2, LS MAX c -CUT presumably cannot be solved in g(k) · n O (1) time even on bipartite graphs. We then show an algorithm for LS MAX c -CUT with running time O((3 e Δ) k · c · k 3 · Δ · n ), where Δ 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 state-of-the-art heuristics for MAX c -CUT. We show that using parameterized local search, the results of this heuristic can be further improved on a set of standard benchmark instances.