Greedy Algorithms for Shortcut Sets and Hopsets

TL;DR

This paper develops and analyzes greedy algorithms for graph metric augmentation objects like shortcut sets and hopsets, achieving near-optimal size and hopbound tradeoffs with improved bounds and new analyses.

Contribution

It introduces a simple greedy algorithm for shortcut sets that matches recent state-of-the-art tradeoffs and provides a new analysis showing near-optimality for hopsets.

Findings

Greedy algorithm for shortcut sets matches recent optimal tradeoffs, up to $O(\log n)$ factors.

Additional preprocessing improves size bounds in some parameter ranges.

The same greedy algorithm is shown to be near-optimal for hopsets, up to $O(\log n)$ factors.

Abstract

For many popular graph metric sparsifiers, such as spanners, emulators, and preservers, simple and elegant greedy algorithms are known that achieve state-of-the-art or existentially optimal tradeoffs between size and quality. The goal of this paper is to develop and analyze comparable greedy algorithms for nearby objects in graph metric augmentation. We show the following: - A simple greedy algorithm for shortcut sets achieves the state-of-the-art size/hopbound tradeoff recently proved by Kogan and Parter (2022), up to $O(\log n)$ factors in the size. Moreover, with an additional preprocessing step, the greedy algorithm subpolynomially improves on the previous size bounds in some range of parameters. - The same greedy algorithm was already known to be existentially optimal for the size/hopbound tradeoff for hopsets, by an analysis of Berman, Raskhodnikova, and Ruan (2010) introduced for transitive-closure spanners. We provide a completely different analysis showing that the algorithm is also existentially optimal (up to $O(\log n)$ factors) for the matching hopset problem, in which one has a budget of roughly $O(m)$ additional edges (for an $m$-edge input graph).