Tight Conditional Lower Bounds for Approximating Diameter in Directed Graphs

Abstract: 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 $m<sup>{2-o(1)}$ 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 $\tilde{O}(m)$ time. Additionally, a line of work concludes with a $3/2$-approximation algorithm for Diameter in weighted directed graphs that runs in $\tilde{O}(m<sup>{3/2})$ time. The $3/2$-approximation algorithm is known to be tight under SETH: Roditty and Vassilevska W. proved that under SETH any $3/2-\epsilon$ approximation algorithm for Diameter in undirected unweighted graphs requires $m<sup>{2-o(1)}$ time, and then Backurs, Roditty, Segal, Vassilevska W., and Wein and the follow-up work of Li proved that under SETH any $5/3-\epsilon$ approximation algorithm for Diameter in undirected unweighted graphs requires $m<sup>{3/2-o(1)}$ 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 $7/4-\epsilon$ approximation requires $m<sup>{4/3-o(1)}$, 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 $\delta&gt;0$, a $(\frac{2k-1}{k}-\delta)$-approximation algorithm for Diameter on directed unweighted graphs requires $m<sup>{\frac{k}{k-1}-o(1)}$ time.