eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-08-23
131:1
131:14
10.4230/LIPIcs.ICALP.2016.131
article
Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds
Cheung, Yun Kuen
Goranci, Gramoz
Henzinger, Monika
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol055-icalp2016/LIPIcs.ICALP.2016.131/LIPIcs.ICALP.2016.131.pdf
Distance Approximating Minor
Graph Minor
Graph Compression
Vertex Sparsification
Metric Embedding