拟度量嵌入及其应用
Quasimetric embeddings and their applications
摘要：
我们研究了经典度量嵌入结果到拟度量空间的推广，即不一定满足对称性的空间。拟度量空间自然产生于有向图上的最短路距离。也许令人惊讶的是，很少有人知道准度量空间的低失真嵌入。超度量空间的随机嵌入可以说是算法设计中最成功的几何工具之一。我们将其推广到拟度量的情况如下。我们证明了树宽$t$图上支持的任何$n$ -点拟度量空间都可以随机嵌入到具有畸变的拟超度量空间$O(t \log^2 n)$中，其中拟超度量是超度量的自然推广.这个结果使我们能够获得$t\log^{O(1)} n$ -近似算法的有向非二分稀疏割和有向多割问题的$n$ -顶点图的树宽$t$，与运行时间多项式在$n$和$t$。上述结果是通过考虑随机划分到拟度量情形的推广而得到的，我们称之为随机拟划分。利用这个定义和[Chuzhoy and Khanna 2009]的一个构造，我们得到了一般拟度量空间到拟超度量空间的随机嵌入的失真的多项式下界。最后，我们建立了一个下界嵌入图$G$的最短路径拟度量到图，排除$G$作为一个小的。这个下界是用来表明，从度量的情况下，几个嵌入结果没有自然的类似物在准度量设置。
英文摘要：
We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces. Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any $n$-point quasimetric space supported on a graph of treewidth $t$ admits a random embedding into quasiultrametric spaces with distortion $O(t \log^2 n)$, where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain $t\log^{O(1)} n$-approximation algorithms for the Directed Non-Bipartite Sparsest-Cut and the Directed Multicut problems on $n$-vertex graphs of treewidth $t$, with running time polynomial in both $n$ and $t$. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of [Chuzhoy and Khanna 2009] we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph $G$ into graphs that exclude $G$ as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.
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