Before revision:  This result improves on lower bounds of Backurs, Roditty, Segal, Vassilevska-Williams, and Wein ) for the Diameter of a sparse directed unweighted graph can be ruled out by a deterministic SETH-based reduction for every \delta\in(0,1) and essentially every \varepsilon\in(0,1), assuming NSETH.
After revision:  This result improves on lower bounds of Backurs, Roditty, Segal, Vassilevska-Williams, and Wein ) time. We show that, assuming a hypothesis called NSETH, one cannot use a deterministic SETH-based reduction to rule out the existence of such algorithms. Furthermore, assuming another hypothesis called Non-Uniform NSETH (NUNSETH), one cannot use a even a randomized SETH-based reduction to rule out the existence of such algorithms. Extending the techniques in these two results, we characterize whether a 2-\varepsilon approximation algorithm running in time O(n^{1+\delta) for the Diameter of a sparse directed unweighted graph can be ruled out by a deterministic SETH-based reduction for every \delta\in(0,1) and essentially every \varepsilon\in(0,1), assuming NSETH.