Better Approximation Algorithms for the Graph Diameter

Better Approximation Algorithms for the Graph Diameter

Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert E. Tarjan, Virginia Vassilevska Williams


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 $\OTilde{n^2+m\sqrt n}$ time an estimate D* for the diameter D in directed graphs with nonnegative edge weights, such that ⌊⅔ · D⌋ – (M – 1) ≤ D* ≤ 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 $\OTilde{m\sqrt n}$. Roditty and Vassilevska W. also showed that unless the Strong Exponential Time Hypothesis fails, no O(n2−) 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(mnn3/2) time. Furthermore, our algorithms are deterministic, and thus we present the first deterministic (2 – )-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 $\OTilde{m\sqrt n}$ time algorithm achieves an additive n-approximation; no better results are known. We show that for any > 0, getting an additive n-approximation algorithm for the diameter running in O (n2−δ) time for any δ > 2 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 $\OTilde{m\sqrt n}$ time, one can compute for each vV in an undirected graph, an estimate ∊(v) for the eccentricity ∊(v) such that max {R, 2/3 · ∊(v)} ≤ ∊(v) ≤ min {D, 3/2 · ∊(v)} where R = minv ∊(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 ∊(v) with 5/3 · ∊(v) ≤ ∊′(v) ≤ ∊(v).


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