Approximation Algorithms and Hardness for $n$-Pairs Shortest Paths and All-Nodes Shortest Cycles

TL;DR

This paper investigates the approximability of the $n$-Pairs Shortest Paths and All Nodes Shortest Cycles problems, providing new algorithms and lower bounds that clarify the trade-offs between running time and approximation quality.

Contribution

It introduces new algorithms with near-linear and subquadratic running times for approximating $n$-PSP and ANSC, and establishes conditional lower bounds based on the $4k$-clique hypothesis.

Findings

Algorithms achieving $ ilde O(m + n^{3/2+ ext{epsilon}})$ time with $2+ ext{epsilon}$ approximation.

Conditional lower bounds showing no better approximation in certain time bounds under the $4k$-clique hypothesis.

Near-linear time algorithms with adjustable trade-offs between speed and accuracy.

Abstract

We study the approximability of two related problems on graphs with $n$ nodes and $m$ edges: $n$-Pairs Shortest Paths ($n$-PSP), where the goal is to find a shortest path between $O(n)$ prespecified pairs, and All Node Shortest Cycles (ANSC), where the goal is to find the shortest cycle passing through each node. Approximate $n$-PSP has been previously studied, mostly in the context of distance oracles. We ask the question of whether approximate $n$-PSP can be solved faster than by using distance oracles or All Pair Shortest Paths (APSP). ANSC has also been studied previously, but only in terms of exact algorithms, rather than approximation. We provide a thorough study of the approximability of $n$-PSP and ANSC, providing a wide array of algorithms and conditional lower bounds that trade off between running time and approximation ratio. A highlight of our conditional lower bounds results is that for any integer $k\ge 1$, under the combinatorial $4k$-clique hypothesis, there is no combinatorial algorithm for unweighted undirected $n$-PSP with approximation ratio better than $1+1/k$ that runs in $O(m^{2-2/(k+1)}n^{1/(k+1)-\epsilon})$ time. This nearly matches an upper bound implied by the result of Agarwal (2014). A highlight of our algorithmic results is that one can solve both $n$-PSP and ANSC in $\tilde O(m+ n^{3/2+\epsilon})$ time with approximation factor $2+\epsilon$ (and additive error that is function of $\epsilon$), for any constant $\epsilon>0$. For $n$-PSP, our conditional lower bounds imply that this approximation ratio is nearly optimal for any subquadratic-time combinatorial algorithm. We further extend these algorithms for $n$-PSP and ANSC to obtain a time/accuracy trade-off that includes near-linear time algorithms.