Finding locally optimal solutions for max-cut and max-k-cut are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin [ER17] showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres and Wei [ABPW17] showed that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ 5 n 15.1 ), where φ is an upper bound on the random edge-weight density and n is the number of vertices in the input graph. While Angel et al.'s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress towards improving the run-time bound. We prove that the smoothed complexity of FLIP in complete graphs is O(φn 7.83 ). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for max-3-cut in complete graphs is polynomial and for max-k-cut in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest towards addressing the smoothed complexity of FLIP for max-k-cut in complete graphs for larger constants k.
Independently, Bibaksereshkeh, Carlson, and Chandrasekaran REF improved the running-time for the local-max-cut algorithm, and obtained smoothed polynomial and quasi-polynomial algorithms for local-3-cut and local-k-cut with constant k respectively.
Improving the smoothed complexity of FLIP for max cut problems
ArXiv
ArXiv
Mathematics Computer Science