This paper considers additive approximation algorithms for All-Pairs Shortest Paths (APSP) and Shortest Cycle in undirected unweighted graphs. The results are as follows: - We obtain the first +2-approximation algorithm for APSP in n-vertex graphs that improves upon Dor, Halperin and Zwick’s (SICOMP'00) Õ(n^{7/3}) time algorithm. The new algorithm runs in Õ(n^2.29) time and is obtained via a reduction to Min-Plus product of bounded difference matrices. - We obtain the first additive approximation scheme for Shortest Cycle, generalizing the approximation algorithms of Itai and Rodeh (SICOMP'78) and Roditty and Vassilevska W. (SODA'12). For every integer r ≥ 0, we give an Õ(n+n^{2+r}/m^r) time algorithm that returns a +(2r+1)-approximate shortest cycle in any n-vertex, m-edge graph.
@InProceedings{deng_et_al:LIPIcs.ICALP.2022.50, author = {Deng, Mingyang and Kirkpatrick, Yael and Rong, Victor and Vassilevska Williams, Virginia and Zhong, Ziqian}, title = {{New Additive Approximations for Shortest Paths and Cycles}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {50:1--50:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.50}, URN = {urn:nbn:de:0030-drops-163919}, doi = {10.4230/LIPIcs.ICALP.2022.50}, annote = {Keywords: Fine-grained Complexity, Additive Approximation} }
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