发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明存在某些DAG，任何保持最短路径的整数重新加权都需要指数级权重，即使结构简单如3层且中间层仅3个顶点，并扩展至近似版本，否定多项式整数边权的可能性。
AI 中文摘要
我们研究了由Bernstein、Bodwin和Wein在[ITCS'24]中引入的一个图简化问题。我们从具有任意大正边权的图出发，目标是在保持最短路径结构（沿最短路径的顶点和边的序列）的同时，将边重新加权为小纵横比（最大权重与最小权重之比）。他们研究了多项式纵横比是否总是可能的。他们证明，对于一般图，无论是有向图还是无向图，答案是否定的：存在某些图，任何保持最短路径的重新加权都需要指数纵横比。相比之下，他们证明了每个DAG（有向无环图）都允许具有线性纵横比的重新加权。然而，由此产生的边权不是整数。这促使他们提出了一个开放性问题：是否所有DAG都允许具有多项式有界整数边权的重新加权。我们的主要结果是对这个问题给出否定答案：我们证明存在某些DAG，任何保持最短路径的整数重新加权都需要大小为$2^{\Omega(n)}$的权重。事实上，即使DAG具有非常简单的结构：3层顶点，中间层只有3个顶点，这一点也成立。相比之下，我们表明如果中间层的顶点数减少到2，则线性上界是可能的。我们将指数下界扩展到问题的近似版本，其中只需保留原始图中的单个$\alpha$-近似最短路径作为重新加权图中的精确最短路径。我们的指数下界甚至对任何有限近似比$\alpha>1$都成立。
英文摘要
We study a graph simplification problem introduced by Bernstein, Bodwin, and Wein [ITCS'24]. We start with a graph with arbitrarily large positive edge weights and the goal is to reweight the edges to small aspect ratio (ratio between largest and smallest weight) while preserving the shortest paths structure (the sequence of vertices and edges along shortest paths). They studied whether polynomial aspect ratio is always possible. They proved that for general graphs, both directed and undirected, it is not: there exist graphs for which any shortest-paths preserving reweighting requires exponential aspect ratio. In contrast, they showed that every DAG (directed acyclic graph) admits a reweighting with linear aspect ratio. However, the resulting edge weights are not integers. This motivated them to pose the open question of whether all DAGs admit a reweighting with polynomially-bounded integer edge weights. Our main result is to answer this question in the negative: we prove that there exist DAGs for which any shortest-paths preserving integer reweighting requires weights of size $2^{Ω(n)}$. In fact, this is even true when the DAG has very simple structure: 3 layers of vertices with only 3 vertices in the middle layer. In contrast, we show that if the number of vertices in the middle layer is decreased to 2, then a linear upper bound is possible. We extend our exponential lower bound to the approximate version of the problem where only a single $α$-approximate shortest path in the original graph must be preserved as an exact shortest path in the reweighted graph. Our exponential lower bound holds even for any finite approximation ratio $α>1$.
