Better approximation algorithms for the graph diameter

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 Õ (n² + m√n) time an estimate D̃ for the diameter D in directed graphs with nonnega- Tive edge weights, such that [2/3 · D]-(M - I) ≤ D̃ ≤ D, 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 Õ (m√;n). Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O(n^2-ε) 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 Õ (m^3/2)time, and one running in Õ (mn ^2/3 ) time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 - ε)-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 D̃ such that D - c ≤ D̃ ≤ D̃. An extremely simple Õ (mn^1-ε) time algorithm achieves an additive n ^ε- Approximation; no better results are known. We show that for any ε > 0, getting an additive n^ε-approximation algorithm for the diameter running in O (n^2-δ) time for any δ > 2ε 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 Õ(m√n) time, one can compute for each v ∈ V in an undirected graph, an estimate e (v) for the eccentricity ε (v) such that max {R, 2/3 · ε (v)} ≤ e (v) ≤ min {D, 3/2 · ε (v)} where R = min_vε (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 ε' (v) with 3/5 · ε (v) ≤ ε' (v) ≤ ε(v).