Finding Sparser Directed Spanners

Abstract

A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, a subgraph $H = (V,E_H)$ is a $k$-spanner of a graph $G=(V,E)$ if for every pair of vertices $u,v \in V$, the shortest path distance  $dist_H(u,v)$ from $u$ to $v$ in $H$ is at most $k.dist_G(u,v)$.  We focus on spanners of directed graphs and a related notion of transitive-closure spanners. The latter captures the idea that a spanner should have a small diameter but preserve the connectivity of the original graph. We study the computational problem of finding the sparsest $k$-spanner (resp., $k$-TC-spanner) of a given directed graph, which we refer to as DIRECTED $k$-SPANNER (resp., $k$-TC-SPANNER). We improve all known approximation algorithms for these problems for $k\geq 3$. (For $k=2$, the current ratios are tight, assuming P$\neq$NP.) Along the way, we prove several structural results about the size of the sparsest spanners of directed graphs.