Tight conditional lower bounds for approximating diameter in directed graphs

Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices in a graph. 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 (where Õ notation suppresses logarithmic factors). Additionally, a line of work [SODA'96, STOC'13, SODA'14] concludes with a 3/2-approximation algorithm for Diameter in weighted directed graphs that runs in Õ(m 3/2 ) time. For directed graphs, these are the only known approximation algorithms for Diameter.

The 3/2-approximation algorithm is known to be tight under SETH: Roditty and Vassilevska W. [STOC'13] 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 [STOC'18] 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.