Settling the Complexity of Local Max-Cut (Almost) Completely
Settling the Complexity of Local Max-Cut (Almost) Completely
复制标题
（几乎）完全解决局部最大割的复杂性
DOI：
10.1007/978-3-642-22006-7_15
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Tobias Tscheuschner
中科院分区：
文献类型：
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作者：
Robert Elsässer;Tobias Tscheuschner
我们考虑了针对Flip-Neighborhood的最大切割问题的局部最佳问题的问题，其中一个节点会改变分区和Yannakakis（Sicomp，1991），显示了该问题的PLS PLS完整性。另一方面，波尔贾克（Sicomp，1995年）表明，在立方图中，每个翻转局部搜索都需要o（n2）步骤，其中n是节点的数量。和Vocking（JACM，2008年）要求最小的D，以便在最大程度上的图形上，局部最大切割问题在本文中是完整的。最大程度的图是PLS完整的，我们通过证明D是四到五的，解决了Ackermann等人的问题。
另一方面，我们还证明，在具有o（log n）的图表上，每个翻转本地搜索都可能具有多项式平滑的复杂性。方差和sigma2，每个翻转本地搜索在n和σ-1中终止于多项式，概率为1-n-ω（1），我们可以包括，尽管局部max-cut可能很难具有有界程度的程度，可以在多项式时间内解决具有很高概率的略微扰动实例。
We consider the problem of finding a local optimum for the Max-Cut problem with FLIP-neighborhood, in which exactly one node changes the partition. Schaffer and Yannakakis (SICOMP, 1991) showed PLS-completeness of this problem on graphs with unbounded degree. On the other side, Poljak (SICOMP, 1995) showed that in cubic graphs every FLIP local search takes O(n2) steps, where n is the number of nodes. Due to the huge gap between degree three and unbounded degree, Ackermann, Roglin, and Vocking (JACM, 2008) asked for the smallest d such that on graphs with maximum degree d the local MAX-CUT problem with FLIP-neighborhood is PLS-complete. In this paper, we prove that the computation of a local optimum on graphs with maximum degree five is PLS-complete. Thus, we solve the problem posed by Ackermann et al. almost completely by showing that d is either four or five (unless PLS ⊆ P).
On the other side, we also prove that on graphs with degree O(log n) every FLIP local search has probably polynomial smoothed complexity. Roughly speaking, for any instance, in which the edge weights are perturbated by a (Gaussian) random noise with variance &sigma2, every FLIP local search terminates in time polynomial in n and σ-1, with probability 1-n-Ω(1). Putting both results together, we may conclude that although local MAX-CUT is likely to be hard on graphs with bounded degree, it can be solved in polynomial time for slightly perturbated instances with high probability.
影响因子：
1.1
作者：
Matthias Englert;Heiko Röglin;Berthold Vöcking
通讯作者：
Matthias Englert;Heiko Röglin;Berthold Vöcking