[论文解读] Are There Graphs Whose Shortest Path Structure Requires Large Edge Weights?

Aaron Bernstein, Greg Bodwin+1 more|arXiv (Cornell University)|Aug 24, 2023

Advanced Graph Theory Research被引用 3

一句话总结

本文研究了具有复杂最短路径结构的图是否可以在不改变其最短路径的前提下，通过重新加权使其具有较低的纵横比（最大边权与最小边权之比）。研究证明，虽然有向无环图（DAG）可以被重新加权以实现纵横比 O(n)，但一般有向图和无向图即使仅近似保留最短路径，也必须具有指数级大的纵横比——2Ω(n)。这为一般图中保持最短路径的纵横比最小化设定了根本性限制。

ABSTRACT

The aspect ratio of a (positively) weighted graph $G$ is the ratio of its maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as a complexity measure in graph algorithms, especially related to the computation of shortest paths. Popular paradigms are to interpolate between the settings of weighted and unweighted input graphs by incurring a dependence on aspect ratio, or by simply restricting attention to input graphs of low aspect ratio. This paper studies the effects of these paradigms, investigating whether graphs of low aspect ratio have more structured shortest paths than graphs in general. In particular, we raise the question of whether one can generally take a graph of large aspect ratio and reweight its edges, to obtain a graph with bounded aspect ratio while preserving the structure of its shortest paths. Our findings are: - Every weighted DAG on $n$ nodes has a shortest-paths preserving graph of aspect ratio $O(n)$. A simple lower bound shows that this is tight. - The previous result does not extend to general directed or undirected graphs; in fact, the answer turns out to be exponential in these settings. In particular, we construct directed and undirected $n$-node graphs for which any shortest-paths preserving graph has aspect ratio $2^{Ω(n)}$. We also consider the approximate version of this problem, where the goal is for shortest paths in $H$ to correspond to approximate shortest paths in $G$. We show that our exponential lower bounds extend even to this setting. We also show that in a closely related model, where approximate shortest paths in $H$ must also correspond to approximate shortest paths in $G$, even DAGs require exponential aspect ratio.

研究动机与目标

• 确定具有高纵横比的图是否可以通过重新加权实现低纵横比，同时保持其最短路径结构。
• 探究有界纵横比图是否本质上比一般图具有更结构化的最短路径。
• 为各种图类在保持最短路径前提下的重新加权所需最小纵横比建立紧致边界。
• 检验这些结果在近似最短路径保持下的鲁棒性。
• 探讨这些发现对最短路径计算中纵横比敏感算法的影响。

提出的方法

• 构建了一类 n 个节点的图，其保持最短路径的重新加权需要指数级大的纵横比。
• 使用基于网格的构造方法，通过精心选择边权，强制仅特定路径可为最短路径。
• 对网格子图应用归纳论证，证明最后一行和最后一列的边权和随 √n 指数增长。
• 采用反证法：假设存在低纵横比的重新加权，进而推导出违反最短路径约束的权值失衡。
• 利用非最短路径必须显著长于最短路径的特性，迫使边权变大。
• 通过证明在 (1+ε)-近似下，指数下界依然成立，将结果扩展至近似最短路径。

实验结果

研究问题

• RQ1 每个 n 个节点的加权有向无环图是否都能被重新加权，使其纵横比为 O(n)，同时保持所有最短路径？
• RQ2 一般有向图或无向图是否允许具有多项式纵横比的、保持最短路径的重新加权？
• RQ3 一般图的纵横比指数下界是否紧致，还是可以改进？
• RQ4 指数下界是否适用于近似最短路径保持？
• RQ5 在近似最短路径约束下，甚至有向无环图是否也可能被迫进入指数级纵横比？

主要发现

• 每个 n 个节点的有向无环图都存在一种保持最短路径的重新加权方式，其纵横比为 O(n)，且该界是紧致的。
• 存在 n 个节点的有向图和无向图，其任何保持最短路径的重新加权都要求纵横比达到 2Ω(n)。
• 即使在保留 (1+ε)-近似最短路径的情况下，该指数下界依然成立。
• 即使在对称模型中要求两个方向的近似路径对应关系，有向无环图仍然需要指数级纵横比。
• 在基于网格的构造中，最后一行和最后一列的边权和增长为 (αH)Ω(√n)，从而至少有一条边的权值达到 (αH)Ω(√n)。
• 这些结果确立了一个根本性障碍：即使近似地，也并非所有最短路径结构都能由低纵横比图捕捉。