Settling the Complexity of Local Max-Cut (Almost) Completely

We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P). On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.