Are there graphs whose shortest path structure requires large edge weights?

TL;DR

This paper investigates whether graphs with large aspect ratios can be reweighted to have bounded aspect ratios while preserving shortest path structures, revealing exponential lower bounds in general graphs.

Contribution

It proves that while DAGs can be reweighted to bounded aspect ratio preserving shortest paths, general directed and undirected graphs require exponential aspect ratios, even approximately.

Findings

DAGs have shortest-paths preserving graphs with aspect ratio O(n)

General directed and undirected graphs can require aspect ratio exponential in n

Exponential lower bounds hold even for approximate shortest path preservation

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 $2^{\Omega(n)}$. 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.