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

08/26/2020

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by   Édouard Bonnet, et al.

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We show, assuming the Strong Exponential Time Hypothesis, that for every ε > 0, approximating directed Diameter on m-arc graphs within ratio 7/4 - ε 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.