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DOI: 10.4230/LIPIcs.ICALP.2016.131
URN: urn:nbn:de:0030-drops-62675
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6267/
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Cheung, Yun Kuen ; Goranci, Gramoz ; Henzinger, Monika

Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds

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LIPIcs-ICALP-2016-131.pdf (0.6 MB)


Abstract

Given a graph where vertices are partitioned into k terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately. The distortion of a compressed graph is the maximum multiplicative blow-up of distances between all pairs of terminals. We study the trade-off between the number of non-terminals and the distortion. This problem generalizes the Steiner Point Removal (SPR) problem, in which all non-terminals must be removed. We introduce a novel black-box reduction to convert any lower bound on distortion for the SPR problem into a super-linear lower bound on the number of non-terminals, with the same distortion, for our problem. This allows us to show that there exist graphs such that every minor with distortion less than 2 / 2.5 / 3 must have Omega(k^2) / Omega(k^{5/4}) / Omega(k^{6/5}) non-terminals, plus more trade-offs in between. The black-box reduction has an interesting consequence: if the tight lower bound on distortion for the SPR problem is super-constant, then allowing any O(k) non-terminals will not help improving the lower bound to a constant. We also build on the existing results on spanners, distance oracles and connected 0-extensions to show a number of upper bounds for general graphs, planar graphs, graphs that exclude a fixed minor and bounded treewidth graphs. Among others, we show that any graph admits a minor with O(log k) distortion and O(k^2) non-terminals, and any planar graph admits a minor with 1 + epsilon distortion and ~O((k/epsilon)^2) non-terminals.

BibTeX - Entry

@InProceedings{cheung_et_al:LIPIcs:2016:6267,
  author =	{Yun Kuen Cheung and Gramoz Goranci and Monika Henzinger},
  title =	{{Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{131:1--131:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6267},
  URN =		{urn:nbn:de:0030-drops-62675},
  doi =		{10.4230/LIPIcs.ICALP.2016.131},
  annote =	{Keywords: Distance Approximating Minor, Graph Minor, Graph Compression, Vertex Sparsification, Metric Embedding}
}

Keywords: Distance Approximating Minor, Graph Minor, Graph Compression, Vertex Sparsification, Metric Embedding
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue Date: 2016
Date of publication: 17.08.2016


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