度量扩张算子、顶点稀疏算子和Lipschitz可扩性
Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability
摘要：
我们研究了最近由Moitra，Leighton和Moitra引入的顶点切割和流稀疏器。我们改进和推广了他们的结果。我们给出了一个新的多项式时间算法来构造O（log k / log log k）的割和流稀疏器，匹配稀疏器质量的最佳存在上界，并改进了以前算法的O（log^2 k / log log k）上界。我们表明，流稀疏可以从线性算子近似最小度量扩展。我们引入（线性）度量扩张算子的概念，证明它们的存在，并给出一个精确的多项式时间算法寻找最优算子。然后，我们建立了一个直接的连接流和削减稀疏和Lipschitz扩张的映射在Banach空间中，一个概念，自20世纪30年代以来在功能分析研究。使用这种连接，我们证明了欧米茄（sqrt{log k/log log k}）流稀疏和欧米茄（sqrt{log k}/log log k）切割稀疏的下界。证明了如果Ball在1992年提出的某个开放问题有肯定的答案，则存在O（sqrt{log k}）的\tilde割稀疏子.另一方面，任何比\tilde Omega（sqrt{log k}）更好的切割稀疏器的下界都意味着这个问题的否定答案。
英文摘要：
We study vertex cut and flow sparsifiers that were recently introduced by Moitra, and Leighton and Moitra. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log^2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we prove a lower bound of Omega(sqrt{log k/log log k}) for flow sparsifiers and a lower bound of Omega(sqrt{log k}/log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist \tilde O(sqrt{log k}) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than \tilde Omega(sqrt{log k}) would imply a negative answer to this question.
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