Inapproximability of Diameter in super-linear time: Beyond the 5/3 ratio

TL;DR

This paper proves, under the Strong Exponential Time Hypothesis, that approximating the directed Diameter within a ratio better than 7/4 in super-linear time is computationally hard, extending the understanding of inapproximability bounds.

Contribution

It establishes the first conditional lower bound ruling out near-linear time 5/3-approximation for directed Diameter, using novel constructions with nonnegative weights.

Findings

Approximating Diameter within 7/4 - ε requires m^{4/3 - o(1)} time.

The results hold for sparse digraphs with m = n log^{O(1)} n.

First conditional inapproximability result beyond 5/3 ratio in super-linear time.

Abstract

We show, assuming the Strong Exponential Time Hypothesis, that for every $\varepsilon > 0$, approximating directed Diameter on $m$-arc graphs within ratio $7/4 - \varepsilon$ requires $m^{4/3 - o(1)}$ time. Our construction uses nonnegative edge weights but even holds for sparse digraphs, i.e., for which the number of vertices $n$ and the number of arcs $m$ satisfy $m = n \log^{O(1)} n$. This is the first result that conditionally rules out a near-linear time $5/3$-approximation for Diameter.