  We prove several tight results on the fine-grained complexity of
approximating the diameter of a graph. First, we prove that, for any
$\varepsilon>0$, assuming the Strong Exponential Time Hypothesis (SETH), there
are no near-linear time $2-\varepsilon$-approximation algorithms for the
Diameter of a sparse directed graph, even in unweighted graphs. This result
shows that a simple near-linear time 2-approximation algorithm for Diameter is
optimal under SETH, answering a question from a survey of Rubinstein and
Vassilevska-Williams (SIGACT '19) for the case of directed graphs.
  In the same survey, Rubinstein and Vassilevska-Williams also asked if it is
possible to show that there are no $2-\varepsilon$ approximation algorithms for
Diameter in a directed graph in $O(n^{1.499})$ 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.
  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. This settles the SETH-hardness of
approximating the diameter of sparse directed unweighted graphs for
deterministic reductions, up to NSETH. We make the same characterization for
randomized SETH-based reductions, assuming another hypothesis called NUNSETH.
  We prove additional hardness and non-reducibility results for undirected
graphs.
