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

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 $1+\varepsilon$ distortion and $\widetilde{O}((k/\varepsilon)^{2})$ non-terminals.