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 non-negative 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 = O˜(n). This is the first result that conditionally rules out a near-linear time 5/3-approximation for a variant of Diameter.
@InProceedings{bonnet:LIPIcs.STACS.2021.17, author = {Bonnet, \'{E}douard}, title = {{Inapproximability of Diameter in Super-Linear Time: Beyond the 5/3 Ratio}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {17:1--17:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.17}, URN = {urn:nbn:de:0030-drops-136623}, doi = {10.4230/LIPIcs.STACS.2021.17}, annote = {Keywords: Diameter, inapproximability, SETH lower bounds, k-Orthogonal Vectors} }
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