平衡路由
Routing under Balance
摘要：
我们引入了有向图平衡的概念：一个加权有向图是$α$ -平衡的，如果对于每一个割$S \subseteq V$，从$S$到$V\setminus S$的边的总权重在从$V\setminus S$到$S$的边的总权重的因子$α$之内.几个重要的图族几乎是平衡的，特别是欧拉图（$α= 1$）和$(1+ε)$ -近似无向最大流（$α=O(1/ε)$）的残差图。我们使用平衡的概念来更细粒度地理解几个研究得很好的路由问题，这些问题在有向图中要困难得多。我们首先回顾有向图中的不经意路由。我们的主要算法的结果是一个不经意的路由方案的单源实例，实现了$O(α\cdot \log^3 n / \log \log n)$竞争比。在这个过程中，我们做出了一些可能具有独立利益的技术贡献。特别是，我们给出了一个有效的算法计算低半径分解的平衡参数化的有向图。我们还定义和构造低拉伸树，低拉伸生成树的有向图的推广。从消极的方面来说，我们为有向图上的不经意路由问题提出了新的下界。我们发现，有向图的不经意路由算法的竞争比是$Ω(n)$一般，这一结果改进了长期以来最知名的$Ω(\sqrt{n})$的下界Hajiaghayi，Kleinberg，Leighton和Räcke在2006年。我们还表明，我们的限制，单源实例是必要的，通过显示一个$Ω(\sqrt{n})$下界多源不经意路由欧拉图。对于平衡有向图中的最大流问题，我们也给出了一个快速算法。
英文摘要：
We introduce the notion of balance for directed graphs: a weighted directed graph is $α$-balanced if for every cut $S \subseteq V$, the total weight of edges going from $S$ to $V\setminus S$ is within factor $α$ of the total weight of edges going from $V\setminus S$ to $S$. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with $α= 1$) and residual graphs of $(1+ε)$-approximate undirected maximum flows (with $α=O(1/ε)$). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an $O(α\cdot \log^3 n / \log \log n)$ competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is $Ω(n)$ in general; this result improves upon the long-standing best known lower bound of $Ω(\sqrt{n})$ given by Hajiaghayi, Kleinberg, Leighton and Räcke in 2006. We also show that our restriction to single-source instances is necessary by showing an $Ω(\sqrt{n})$ lower bound for multiple-source oblivious routing in Eulerian graphs. We also give a fast algorithm for the maximum flow problem in balanced directed graphs.
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