Better approximation algorithms for the graph diameter

Abstract

The diameter is a fundamental graph parameter and its computation is necessary in many applications. The fastest known way to compute the diameter exactly is to solve the All-Pairs Shortest Paths (APSP) problem.In the absence of fast algorithms, attempts were made to seek fast algorithms that approximate the diameter. In a seminal result Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] designed an algorithm that computes in O (n2 + m√n) time an estimate D for the diameter D in directed graphs with nonnegative edge weights, such that [EQUATION], where M is the maximum edge weight in the graph. In recent work, Roditty and Vassilevska W. [STOC 13] gave a Las Vegas algorithm that has the same approximation guarantee but improves the (expected) runtime to O (m√n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O (n2-e) time algorithm for sparse unweighted undirected graphs can achieve an approximation ratio better than 3/2. Thus their algorithm is essentially tight for sparse unweighted graphs. For weighted graphs however, the approximation guarantee can be meaningless, as M can be arbitrarily large.In this paper we exhibit two algorithms that achieve a genuine 3/2-approximation for the diameter, one running in O (m3/2) time, and one running in O (mn2/3). time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 -- e)-approximation algorithm for the diameter that takes subquadratic time in sparse graphs.In addition, we address the question of obtaining an additive c-approximation for the diameter, i.e. an estimate D such that D -- c ≤ D ≤ D. An extremely simple O (mn1-e) time algorithm achieves an additive ne-approximation; no better results are known. We show that for any e > 0, getting an additive ne-approximation algorithm for the diameter running in O (n2-e) time for any δ > 2e would falsify the Strong Exponential Time Hypothesis. Thus the simple algorithm is probably essentially tight for sparse graphs, and moreover, obtaining a subquadratic time additive c-approximation for any constant c is unlikely.Finally, we consider the problem of computing the eccentricities of all vertices in an undirected graph, i.e. the largest distance from each vertex. Roditty and Vassilevska W. [STOC 13] show that in O (m√n) time, one can compute for each v e V in an undirected graph, an estimate e(v) for the eccentricity e (v) such that max{R, 2/3 · e(v)} ≤ e (v) ≤ min {D, 3/2 · e(v)} where R = minv e (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates e' (v) with 3/5 · e (v) ≤ e' (v) ≤ e (v).