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Retracting Graphs to Cycles

Authors Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, Ravi Sundaram



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Author Details

Samuel Haney
  • Duke University, Durham, NC, USA
Mehraneh Liaee
  • Northeastern University, Boston, MA, USA
Bruce M. Maggs
  • Duke University, Durham, NC, USA
  • Akamai Technologies, Cambridge, MA, USA
Debmalya Panigrahi
  • Duke University, Durham, NC, USA
Rajmohan Rajaraman
  • Northeastern University, Boston, MA, USA
Ravi Sundaram
  • Northeastern University, Boston, MA, USA

Acknowledgements

We would like to thank Seffi Naor for helpful discussions on the problems considered in this paper.

Cite AsGet BibTex

Samuel Haney, Mehraneh Liaee, Bruce M. Maggs, Debmalya Panigrahi, Rajmohan Rajaraman, and Ravi Sundaram. Retracting Graphs to Cycles. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 70:1-70:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.70

Abstract

We initiate the algorithmic study of retracting a graph into a cycle in the graph, which seeks a mapping of the graph vertices to the cycle vertices so as to minimize the maximum stretch of any edge, subject to the constraint that the restriction of the mapping to the cycle is the identity map. This problem has its roots in the rich theory of retraction of topological spaces, and has strong ties to well-studied metric embedding problems such as minimum bandwidth and 0-extension. Our first result is an O(min{k, sqrt{n}})-approximation for retracting any graph on n nodes to a cycle with k nodes. We also show a surprising connection to Sperner’s Lemma that rules out the possibility of improving this result using certain natural convex relaxations of the problem. Nevertheless, if the problem is restricted to planar graphs, we show that we can overcome these integrality gaps by giving an optimal combinatorial algorithm, which is the technical centerpiece of the paper. Building on our planar graph algorithm, we also obtain a constant-factor approximation algorithm for retraction of points in the Euclidean plane to a uniform cycle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Graph algorithms
  • Graph embedding
  • Planar graphs
  • Approximation algorithms

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