Before revision: We prove that , assuming the Strong Exponential Time Hypothesis , for any \delta>0, a 5/3-\delta approximation of the diameter of an undirected unweighted graph with nvertices needs n ^{ 3/2-o(1) time.
After revision: We prove that , 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 ^{ 3/2-o(1) time.