@inproceedings{11836,
  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.},
  author       = {Cheung, Yun Kuen and Goranci, Gramoz and Henzinger, Monika H},
  booktitle    = {43rd International Colloquium on Automata, Languages, and Programming},
  isbn         = {978-3-95977-013-2},
  issn         = {1868-8969},
  location     = {Rome, Italy},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Graph minors for preserving terminal distances approximately - lower and upper bounds}},
  doi          = {10.4230/LIPICS.ICALP.2016.131},
  volume       = {55},
  year         = {2016},
}