Tight Conditional Lower Bounds for Approximating Diameter in Directed Graphs
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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 m^2-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 Õ(m) time. Additionally, a line of work concludes with a 3/2-approximation algorithm for Diameter in weighted directed graphs that runs in Õ(m^3/2) 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 m^2-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 m^3/2-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 m^4/3-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.
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