Improving the Smoothed Complexity of FLIP for Max Cut Problems

Ali Bibak, Charles Carlson, Karthekeyan Chandrasekaran

Research output: Contribution to journalArticlepeer-review

Abstract

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 (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is (Oφ5n15.1), where φ is an upper bound on the random edge-weight density and φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei'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 toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O(φ n7.83). 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 toward showing smoothed polynomial complexity of FLIP for MAX-k-CUT in complete graphs for larger constants k.

Original languageEnglish (US)
Article number19
JournalACM Transactions on Algorithms
Volume17
Issue number3
DOIs
StatePublished - Aug 2021

Keywords

  • k-cut
  • local search
  • max-cut
  • Smoothed analysis

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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