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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We consider the problem of listing all avoidable vertices in a given n vertex graph. A vertex is avoidable if every pair of its neighbors is connected by a path whose internal vertices are not neighbors of the vertex or the vertex itself. Recently, Papadopolous and Zisis showed that one can list all avoidable vertices in O(n^{ω+1}) time, where ω < 2.373 is the square matrix multiplication exponent, and conjectured that a faster algorithm is not possible.
In this paper we show that under the 3-OV Hypothesis, and thus the Strong Exponential Time Hypothesis, n^{3-o(1)} time is needed to list all avoidable vertices, and thus the current best algorithm is conditionally optimal if ω = 2. We then show that if ω > 2, one can obtain an improved algorithm that for the current value of ω runs in O(n^3.32) time. We also show that our conditional lower bound is actually higher and supercubic, under a natural High Dimensional 3-OV hypothesis, implying that for our current knowledge of rectangular matrix multiplication, the avoidable vertex listing problem likely requires Ω(n^3.25) time. We obtain further algorithmic improvements for sparse graphs and bounded degree graphs.

Mingyang Deng, Virginia Vassilevska Williams, and Ziqian Zhong. New Lower Bounds and Upper Bounds for Listing Avoidable Vertices. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 41:1-41:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{deng_et_al:LIPIcs.MFCS.2022.41, author = {Deng, Mingyang and Vassilevska Williams, Virginia and Zhong, Ziqian}, title = {{New Lower Bounds and Upper Bounds for Listing Avoidable Vertices}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.41}, URN = {urn:nbn:de:0030-drops-168392}, doi = {10.4230/LIPIcs.MFCS.2022.41}, annote = {Keywords: Avoidable Vertex, Fine-Grained Complexity} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

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.

Mingyang Deng, Yael Kirkpatrick, Victor Rong, Virginia Vassilevska Williams, and Ziqian Zhong. New Additive Approximations for Shortest Paths and Cycles. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 50:1-50:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@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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