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DOI: 10.4230/LIPIcs.ITCS.2024.38

Graph Threading

Authors: Erik D. Demaine, Yael Kirkpatrick, and Rebecca Lin

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

Inspired by artistic practices such as beadwork and himmeli, we study the problem of threading a single string through a set of tubes, so that pulling the string forms a desired graph. More precisely, given a connected graph (where edges represent tubes and vertices represent junctions where they meet), we give a polynomial-time algorithm to find a minimum-length closed walk (representing a threading of string) that induces a connected graph of string at every junction. The algorithm is based on a surprising reduction to minimum-weight perfect matching. Along the way, we give tight worst-case bounds on the length of the optimal threading and on the maximum number of times this threading can visit a single edge. We also give more efficient solutions to two special cases: cubic graphs and the case when each edge can be visited at most twice.

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Erik D. Demaine, Yael Kirkpatrick, and Rebecca Lin. Graph Threading. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{demaine_et_al:LIPIcs.ITCS.2024.38,
  author =	{Demaine, Erik D. and Kirkpatrick, Yael and Lin, Rebecca},
  title =	{{Graph Threading}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{38:1--38:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.38},
  URN =		{urn:nbn:de:0030-drops-195665},
  doi =		{10.4230/LIPIcs.ITCS.2024.38},
  annote =	{Keywords: Shortest walk, Eulerian cycle, perfect matching, beading}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2022.50

New Additive Approximations for Shortest Paths and Cycles

Authors: Mingyang Deng, Yael Kirkpatrick, Victor Rong, Virginia Vassilevska Williams, and Ziqian Zhong

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

Abstract

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.

Cite as

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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Graph Threading

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New Additive Approximations for Shortest Paths and Cycles

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