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- W4289753361 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 R{o}glin showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei showed that the smoothed complexity of FLIP for max-cut in complete graphs is $O(phi^5n^{15.1})$, where $phi$ 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 $O(phi n^{7.83})$. 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 $k$." @default.
- W4289753361 created "2022-08-04" @default.
- W4289753361 creator A5011591780 @default.
- W4289753361 creator A5061561940 @default.
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- W4289753361 date "2018-07-15" @default.
- W4289753361 modified "2023-10-14" @default.
- W4289753361 title "Improving the smoothed complexity of FLIP for max cut problems" @default.
- W4289753361 doi "https://doi.org/10.48550/arxiv.1807.05665" @default.
- W4289753361 hasPublicationYear "2018" @default.
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