On the Power of Nodes of Degree Four in the Local Max-Cut Problem

We study the complexity of local search in the max-cut problem with FLIP neighborhood, in which exactly one node changes the partition. We introduce a technique of constructing instances which enforce certain sequences of improving steps. Using our technique we can show that already graphs with maximum degree four satify the following two properties.There are instances with initial solutions for which every local search takes exponential time to converge to a local optimum. The problem of computing a local optimum reachable from a given solution by a sequence of improving steps is PSPACE-complete. Schaffer and Yannakakis (JOC '91) showed via a so called “tight” PLS-reduction that the properties (1) and (2) hold for graphs with unbounded degree. Our improvement to the degree four is the best possible improvement since Poljak (JOC '95) showed for cubic graphs that every sequence of improving steps has polynomial length, whereby his result is easily generalizable to arbitrary graphs with maximum degree three. In his paper Poljak also asked whether (1) holds for graphs with maximum degree four, which is settled by our result. Many tight PLS-reductions in the literature are based on the max-cut problem. Via some of them our constructions carry over to other problems and show that the properties (1) and (2) already hold for very restricted sets of feasible inputs of these problems. Since our paper provides the two results that typically come along with tight PLS-reductions it does naturally put the focus on the question whether it is even PLS-complete to compute a local optimum on graphs with maximum degree four – a question that was recently asked by Ackermann et al. We think that our insights might be helpful for tackling this question.