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 [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(φ5n15.1), 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(φn7.83). 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 showing smoothed polynomial complexity of FLIP for max-k-cut in complete graphs for larger constants k.
Original language | English (US) |
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Pages | 897-916 |
Number of pages | 20 |
DOIs | |
State | Published - 2019 |
Event | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: Jan 6 2019 → Jan 9 2019 |
Conference
Conference | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 |
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Country/Territory | United States |
City | San Diego |
Period | 1/6/19 → 1/9/19 |
ASJC Scopus subject areas
- Software
- General Mathematics