4 vs 7 sparse undirected unweighted Diameter is SETH-hard at time $n^{4/3}$

TL;DR

This paper proves, under the Strong Exponential Time Hypothesis, that approximating the diameter of certain graphs within a specific ratio requires at least $n^{4/3}$ time, ruling out faster algorithms for near-linear time approximations.

Contribution

It establishes the first conditional lower bound for near-linear time approximation of undirected Diameter within a 7/4 - ε ratio.

Findings

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

First conditional hardness result for near-linear time 5/3-approximation.

Supports the conjecture that faster algorithms for Diameter approximation are unlikely.

Abstract

We show, assuming the Strong Exponential Time Hypothesis, that for every $\varepsilon > 0$, approximating undirected unweighted Diameter on $n$-vertex $n^{1+o(1)}$-edge graphs within ratio $7/4 - \varepsilon$ requires $n^{4/3 - o(1)}$ time. This is the first result that conditionally rules out a near-linear time $5/3$-approximation for undirected Diameter.