All-Hops Shortest Paths

TL;DR

This paper studies algorithms for computing shortest paths with a limited number of edges (hops) in weighted directed graphs, covering various problem variants and providing both algorithms and lower bounds.

Contribution

It introduces new algorithms and bounds for computing hop-constrained shortest paths in different graph settings and for various problem versions.

Findings

Developed algorithms for all-hops shortest paths in weighted graphs.

Established matching conditional lower bounds for some problem variants.

Analyzed both arbitrary and small integer edge weights.

Abstract

Let $G=(V,E,w)$ be a weighted directed graph without negative cycles. For two vertices $s,t\in V$, we let $d_{\le h}(s,t)$ be the minimum, according to the weight function $w$, of a path from $s$ to $t$ that uses at most $h$ edges, or hops. We consider algorithms for computing $d_{\le h}(s,t)$ for every $1\le h\le n$, where $n=|V|$, in various settings. We consider the single-pair, single-source and all-pairs versions of the problem. We also consider a distance oracle version of the problem in which we are not required to explicitly compute all distances $d_{\le h}(s,t)$, but rather return each one of these distances upon request. We consider both the case in which the edge weights are arbitrary, and in which they are small integers in the range $\{-M,\ldots,M\}$. For some of our results we obtain matching conditional lower bounds.