Settling the complexity of local max-cut (almost) completely

TL;DR

This paper determines the complexity threshold for local Max-Cut problems on graphs with bounded degree, proving PLS-completeness for degree five and showing polynomial smoothed complexity for graphs with degree O(log n).

Contribution

It establishes that local Max-Cut is PLS-complete on graphs with maximum degree five, nearly resolving a long-standing open problem, and demonstrates polynomial smoothed complexity for graphs with degree O(log n).

Findings

PLS-complete for degree five graphs.

Polynomial smoothed complexity for graphs with degree O(log n).

Local Max-Cut likely hard on bounded degree graphs, but efficiently solvable with perturbations.

Abstract

We consider the problem of finding a local optimum for Max-Cut with FLIP-neighborhood, in which exactly one node changes the partition. Schaeffer 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(n^2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roeglin, and Voecking (JACM, 2008) asked for the smallest d for which the local Max-Cut problem with FLIP-neighborhood on graphs with maximum degree d 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 is in 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 \sigma^2, every FLIP local search terminates in time polynomial in n and \sigma^{-1}, with probability 1-n^{-\Omega(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.