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 ( O Φ 5 n 15.1 ), 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 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 toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k .

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针对最大切割问题提高 FLIP 的平滑复杂度

寻找局部最优解最大切和最大限度-ķ-切是众所周知的 PLS 完全问题。寻找这种局部最优解的本能方法是 FLIP 方法。尽管 FLIP 在最坏的情况下需要指数级的时间，但在实际情况下它往往会很快终止。为了解释这种差异，我们在平滑复杂度框架中研究了 FLIP 的运行时间。Etscheid 和 Röglin（ACM Transactions on Algorithms，2017）表明 FLIP 的平滑复杂性对于最大切在任意图中是拟多项式。Angel、Bubeck、Peres 和 Wei (STOC, 2017) 表明 FLIP 的平滑复杂性对于最大切在完整的图中是（○Φ5 n 15.1)，其中 Φ 是随机边权重密度的上限，Φ 是输入图中的顶点数。虽然 Angel、Bubeck、Peres 和 Wei 的结果显示了第一个多项式平滑复杂度，但他们也推测他们的运行时界限远非最优。在这项工作中，我们在改进运行时界限方面取得了实质性进展。我们证明了 FLIP 的平滑复杂度最大切在完整的图表中是○(Φn 7.83）。我们的结果基于精心选择的矩阵，其秩捕获了该方法的运行时间，同时改进了该矩阵的秩边界和基于该矩阵的改进的联合边界。此外，我们的技术为在平滑框架中分析 FLIP 提供了一个通用框架。我们通过展示 FLIP 的平滑复杂度来说明这个一般框架MAX-3-CUT在完全图中是多项式并且对于最大限度-ķ-切在任意图中是拟多项式。我们相信我们的技术也应该对展示 FLIP 的平滑多项式复杂性感兴趣最大限度-ķ-切在更大常数的完整图中ķ.