Improved Approximation for the Directed Spanner Problem

Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, Grigory Yaroslavtsev

Abstract: We give an $O(\sqrt{n}\log n)$ -approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V,E) with nonnegative edge lengths d: E → ℝ ≥ 0 and a stretch k ≥ 1, a subgraph H = (V,E H ) is a k-spanner of G if for every edge (u,v) ∈ E, the graph H contains a path from u to v of length at most k ·d(u,v). The previous best approximation ratio was $\tilde{O}(n^{2/3})$ , due to Dinitz and Krauthgamer (STOC ’11). We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is $\tilde{O}(n^{1/3})$ which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS ’10) and Dinitz and Krauthgamer, had approximation ratio $\tilde{O}(\sqrt{n})$ .