2020/11/08 by Dalirrooyfard, Mina, Wein, Nicole · 1 citation
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2011.03892
Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices. Computing the Diameter of a graph with m edges requires m2-o(1) time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a 2-approximation for Diameter in O(m) time. Additionally, a line of work concludes with a 3/2-approximation algorithm for Diameter in weighted directed graphs that runs in O(m3/2) time. The 3/2-approximation algorithm is known to be tight under SETH: Roditty and Vassilevska W. proved that under SETH any 3/2-ε approximation algorithm for Diameter in undirected unweighted graphs requires m2-o(1) time, and then Backurs, Roditty, Segal, Vassilevska W., and Wein and the follow-up work of Li proved that under SETH any 5/3-ε approximation algorithm for Diameter in undirected unweighted graphs requires m3/2-o(1) time. Whether or not the folklore 2-approximation algorithm is tight, however, is unknown, and has been explicitly posed as an open problem in numerous papers. Towards this question, Bonnet recently proved that under SETH, any 7/4-ε approximation requires m4/3-o(1), only for directed weighted graphs. We completely resolve this question for directed graphs by proving that the folklore 2-approximation algorithm is conditionally optimal. In doing so, we obtain a series of conditional lower bounds that together with prior work, give a complete time-accuracy trade-off that is tight with all known algorithms for directed graphs. Specifically, we prove that under SETH for any δ>0, a ((2k-1)/(k)-δ)-approximation algorithm for Diameter on directed unweighted graphs requires m(k)/(k-1)-o(1) time.