[论文解读] Flow-cut gaps and face covers in planar graphs

Robert Krauthgamer, James R. Lee+1 more|arXiv (Cornell University)|Jan 6, 2019

Advanced Graph Theory Research被引用 6

一句话总结

该论文证明了在终端位于γ个面的平面网络中，流-割比间隙为O(log γ)，显著优于先前的界。通过证明终端上的最短路径度量可随机嵌入到树中且畸变度为O(log γ)，作者获得了紧致的渐近界，从而解决了平面图网络流理论中长期存在的开放问题。

ABSTRACT

The relationship between the sparsest cut and the maximum concurrent multi-flow in graphs has been studied extensively. For general graphs, the worst-case gap between these two quantities is now settled: When there are k terminal pairs, the flow-cut gap is O(log k), and this is tight. But when topological restrictions are placed on the flow network, the situation is far less clear. In particular, it has been conjectured that the flow-cut gap in planar networks is O(1), while the known bounds place the gap somewhere between 2 (Lee and Raghavendra, 2003) and [MATH HERE] (Rao, 1999).A seminal result of Okamura and Seymour (1981) shows that when all the terminals of a planar network lie on a single face, the flow-cut gap is exactly 1. This setting can be generalized by considering planar networks where the terminals lie on one of γ > 1 faces in some fixed planar drawing. Lee and Sidiropoulos (2009) proved that the flow-cut gap is bounded by a function of γ and Chekuri, Shepherd, and Weibel (2013) showed that the gap is at most 3γ. We significantly improve these asymptotics by establishing that the flow-cut gap is O(log γ). This is achieved by showing that the edge-weighted shortest-path metric induced on the terminals admits a stochastic embedding into trees with distortion O(log γ). The latter result is tight, e.g., for a square planar lattice on Θ(γ) vertices.The preceding results refer to the setting of edge-capacitated networks. For vertex-capacitated networks, it can be significantly more challenging to control flow-cut gaps. While there is no exact vertex-capacitated version of the Okamura-Seymour Theorem, an approximate version holds; Lee, Mendel, and Moharrami (2015)

研究动机与目标

• 弥合已知的流-割比间隙上界与下界在终端位于多个面的平面网络中的差距。
• 将Okamura-Seymour定理的适用范围从单一面终端配置推广至更一般情形。
• 在γ个面终端约束下，建立平面图中流-割比间隙的紧致渐近界。
• 开发一种保持最短路径度量、畸变度为O(log γ)的随机嵌入技术。
• 将结果从边容量网络推广至顶点容量网络，其中流-割比间隙更为复杂。

提出的方法

• 证明在γ个面平面网络中，终端上的边加权最短路径度量可随机嵌入到树中，且期望畸变度为O(log γ)。
• 利用概率方法和随机划分构造具有可控畸变度的随机嵌入。
• 利用平面图的结构特性和面覆盖性质，界定嵌入的畸变度。
• 通过已知的树度量与多商品流之间的关系，应用嵌入结果来上界流-割比间隙。
• 通过将嵌入框架适配至顶点容量，将分析扩展至顶点容量网络。
• 以现有近似顶点容量Okamura-Seymour定理结果为基础，构建顶点容量情形的扩展分析。

实验结果

研究问题

• RQ1 在终端位于γ个面的平面网络中，流-割比间隙的最紧致渐近界是什么？
• RQ2 能否在γ个面平面网络中，将终端上的最短路径度量随机嵌入到树中，且畸变度为O(log γ)？
• RQ3 平面图中流-割比间隙如何随终端面数γ变化？是否可超越先前的界？
• RQ4 Okamura-Seymour定理在多大程度上可推广至顶点容量网络？
• RQ5 是否存在一个通用框架，通过度量嵌入来界定拓扑受限图中的流-割比间隙？

主要发现

• 在终端位于γ个面的平面网络中，流-割比间隙为O(log γ)，优于先前的3γ界。
• 此类网络中终端上的最短路径度量可实现随机嵌入至树中，且期望畸变度为O(log γ)，对某些平面格点而言该界是紧致的。
• 该结果建立了近乎最优的渐近界，接近平面网络中O(1)间隙的猜想。
• 随机嵌入技术为在平面设置中通过树度量关联流值与割值提供了一般性方法。
• 该框架可扩展至顶点容量网络，其中使用近似版Okamura-Seymour定理推导出界。
• O(log γ)界在常数因子意义下是紧致的，如Θ(γ)顶点的方形平面格点构造所示。