用演化集局部寻找稀疏割
Finding Sparse Cuts Locally Using Evolving Sets
摘要：
{ \em局部图分割算法}通过从指定顶点开始自适应地探索大图$G$的一部分来找到具有小传导性（即，稀疏切割）的顶点的集合。为了使算法是局部的，其复杂性必须以其输出的集合的大小为界，至多弱依赖于$G$中顶点的数目$n$。先前的局部分割算法使用随机游走和个性化PageRank来找到稀疏切割。在本文中，我们介绍了一个随机的局部分割算法，找到一个稀疏切割模拟{ \em体积偏置的进化集过程}，这是一个马尔可夫链的顶点集。我们证明，对于任何一组顶点$A$，具有电导在最$ϕ$，至少有一半的起始顶点在$A$我们的算法将输出（概率至少一半），一组电导$O(ϕ^{1/2} \log^{1/2} n)$。我们证明，对于一个给定的运行的算法，其计算复杂性和它输出的集合的体积之间的预期比率是$O(ϕ^{-1/2} polylog(n))$。相比之下，Andersen、Chung和Lang提出的最佳局部分割算法具有相同的近似保证，但复杂度和输出量之间的$O(ϕ^{-1} polylog(n))$比率更大。使用我们的局部分割算法作为一个子程序，我们构造了一个快速算法寻找平衡削减。给定$ϕ$的固定值，所得到的算法具有复杂度$O((m+nϕ^{-1/2}) polylog(n))$，并且返回具有电导$O(ϕ^{1/2} \log^{1/2} n)$和体积至少为$v_ϕ/2$的切割，其中$v_ϕ$是电导至多为$ϕ$的任何集合中的最大体积。
英文摘要：
A {\em local graph partitioning algorithm} finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph $G$, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number $n$ of vertices in $G$. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the {\em volume-biased evolving set process}, which is a Markov chain on sets of vertices. We prove that for any set of vertices $A$ that has conductance at most $ϕ$, for at least half of the starting vertices in $A$ our algorithm will output (with probability at least half), a set of conductance $O(ϕ^{1/2} \log^{1/2} n)$. We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is $O(ϕ^{-1/2} polylog(n))$. In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of $O(ϕ^{-1} polylog(n))$ between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of $ϕ$, the resulting algorithm has complexity $O((m+nϕ^{-1/2}) polylog(n))$ and returns a cut with conductance $O(ϕ^{1/2} \log^{1/2} n)$ and volume at least $v_ϕ/2$, where $v_ϕ$ is the largest volume of any set with conductance at most $ϕ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
