Graph minors for preserving terminal distances approximately - lower and upper bounds

Cheung YK, Goranci G, Henzinger MH. 2016. Graph minors for preserving terminal distances approximately - lower and upper bounds. 43rd International Colloquium on Automata, Languages, and Programming. ICALP: International Colloquium on Automata, Languages, and Programming, LIPIcs, vol. 55, 131.

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Author
Cheung, Yun Kuen; Goranci, Gramoz; Henzinger, MonikaISTA
Series Title
LIPIcs
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.
Publishing Year
Date Published
2016-08-23
Proceedings Title
43rd International Colloquium on Automata, Languages, and Programming
Publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
Volume
55
Article Number
131
Conference
ICALP: International Colloquium on Automata, Languages, and Programming
Conference Location
Rome, Italy
Conference Date
2016-07-12 – 2016-07-15
ISSN
IST-REx-ID

Cite this

Cheung YK, Goranci G, Henzinger MH. Graph minors for preserving terminal distances approximately - lower and upper bounds. In: 43rd International Colloquium on Automata, Languages, and Programming. Vol 55. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2016. doi:10.4230/LIPICS.ICALP.2016.131
Cheung, Y. K., Goranci, G., & Henzinger, M. H. (2016). Graph minors for preserving terminal distances approximately - lower and upper bounds. In 43rd International Colloquium on Automata, Languages, and Programming (Vol. 55). Rome, Italy: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPICS.ICALP.2016.131
Cheung, Yun Kuen, Gramoz Goranci, and Monika H Henzinger. “Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds.” In 43rd International Colloquium on Automata, Languages, and Programming, Vol. 55. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. https://doi.org/10.4230/LIPICS.ICALP.2016.131.
Y. K. Cheung, G. Goranci, and M. H. Henzinger, “Graph minors for preserving terminal distances approximately - lower and upper bounds,” in 43rd International Colloquium on Automata, Languages, and Programming, Rome, Italy, 2016, vol. 55.
Cheung YK, Goranci G, Henzinger MH. 2016. Graph minors for preserving terminal distances approximately - lower and upper bounds. 43rd International Colloquium on Automata, Languages, and Programming. ICALP: International Colloquium on Automata, Languages, and Programming, LIPIcs, vol. 55, 131.
Cheung, Yun Kuen, et al. “Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds.” 43rd International Colloquium on Automata, Languages, and Programming, vol. 55, 131, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016, doi:10.4230/LIPICS.ICALP.2016.131.
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