Fast approximation algorithms for the diameter and radius of sparse graphs

The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of Õ(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] presented an algorithm that computes in Õ (m √ n+n²) time an estimate D̂ for the diameter D, such that ⌊2/3D⌋ ≤ D̂ ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years. Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et al. , producing an algorithm with the same estimate but with an expected running time of Õ(m √ n). We thus show that for all sparse enough graphs, the diameter can be 3/2-approximated in o(n²) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node. We also provide strong evidence that our diameter approximation result may be hard to improve. We show that if for some constant ε > 0 there is an O(m^2-ε) time (3/2 - ε)- approximation algorithm for the diameter of undirected unweighted graphs, then there is an O * ((2 - δ)ⁿ) time algorithm for CNF-SAT on n variables for constant δ > 0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false. Motivated by this negative result, we give several improved diameter approximation algorithms for special cases. We show for instance that for unweighted graphs of constant diameter D not divisible by 3, there is an O(m^2-ε) time algorithm that gives a (3/2 - ε) approximation for constant ε > 0. This is interesting since the diameter approximation problem is hardest to solve for small D.