Reachability Preservers: New Extremal Bounds and Approximation Algorithms

TL;DR

This paper introduces new bounds on the size of reachability preservers in directed graphs and connects extremal graph sparsification with Steiner Network problems, leading to improved approximation algorithms.

Contribution

It provides the first extremal bounds for reachability preservers and establishes a novel link to Steiner Network problems, improving approximation ratios.

Findings

Reachability preservers can be as small as O(n + √(n|P||S|)) edges.

A lower bound construction shows the bounds are tight in many cases.

Approximation algorithms for Steiner problems improved from O(n^{0.6+ε}) to O(n^{4/7+ε}).

Abstract

We abstract and study \emph{reachability preservers}, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph $G = (V, E)$ and a set of \emph{demand pairs} $P \subseteq V \times V$, a reachability preserver is a sparse subgraph $H$ that preserves reachability between all demand pairs. Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an $n$-node graph and demand pairs of the form $P \subseteq S \times V$ for a small node subset $S$, there is always a reachability preserver on $O(n+\sqrt{n |P| |S|})$ edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which $O(n)$ size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the $O(n^{0.6+\varepsilon})$ of Chlamatac, Dinitz, Kortsarz, and Laekhanukit (SODA'17) to $O(n^{4/7+\varepsilon})$.