In the Steiner point removal (SPR) problem, we are given a weighted graph @math and a set of terminals @math of size @math . The objective is to find a minor @math of @math with only the terminals as its vertex set, such that the distance between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a ball-growing algorithm with exponential distributions to show that the distortion is at most @math . Cheung [Che17] improved the analysis of the same algorithm, bounding the distortion by @math . We improve the analysis of this ball-growing algorithm even further, bounding the distortion by @math .
Krauthgamer, Nguyen and Zondiner @cite_10 showed that if we allowing the minor @math to contain at most @math Steiner vertices in addition to the terminals, then distortion @math can be achieved. They further showed that for graphs with constant treewidth, @math Steiner points will suffice for distortion @math . Cheung, Gramoz and Henzinger @cite_3 showed that allowing @math Steiner vertices, one can achieve distortion @math (in particular distortion @math with @math Steiners). For planar graphs, Cheung et. al. achieved @math distortion with @math Steiner points.
@cite_10 @cite_3
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We introduce the following notion of compressing an undirected graph @math with (nonnegative) edge-lengths and terminal vertices @math . A distance-preserving minor is a minor @math (of @math ) with possibly different edge-lengths, such that @math and the shortest-path distance between every pair of terminals is exactly the same in @math and in @math . We ask: what is the smallest @math such that every graph @math with @math terminals admits a distance-preserving minor @math with at most @math vertices? Simple analysis shows that @math . Our main result proves that @math , significantly improving on the trivial @math . Our lower bound holds even for planar graphs @math , in contrast to graphs @math of constant treewidth, for which we prove that @math vertices suffice. 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.