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 additionally 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 LS-Max c-Cut presumably cannot be solved in g(k) · nᴼ⁽¹⁾ time even on bipartite graphs, for all c ≥ 2. We then show an algorithm for LS-Max c-Cut with running time O((3eΔ)ᵏ · c · k³ · 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.