Cutting corners cheaply, or how to remove Steiner points

Our main result is that the Steiner point removal (SPR) problem can always be solved with polylogarithmic distortion, which answers in the affirmative a question posed by Chan, Xia, Konjevod, and Richa in 2006. Specifically, we prove that for every edge-weighted graph G = (V, E, w) and a subset of terminals T ⊆ V , there is a graph G' = (T, E', w') that is isomorphic to a minor of G such that for every two terminals u, v ∈ T, the shortest-path distances between them in G and in G' satisfy d_G,w (u, v) ≤ d_G',w' (u, v) ≤ O(log⁵|T| · d_G,w (u, v). Our existence proof actually gives a randomized polynomial-time algorithm. Our proof features a new variant of metric decomposition. It is well known that every finite metric space (X, d) admits a β-separating decomposition for β = O(log|X|), which means that for every Δ > 0 there is a randomized partitioning of X into clusters of diameter at most Δ, satisfying the following separation property: for every x, y ∈ X, the probability that they lie in different clusters of the partition is at most β d(x, y)/Δ. We introduce an additional requirement in the form of a tail bound: for every shortest-path P of length d(P) < Δ/β, the number of clusters of the partition that meet the path P, denoted by Z_P, satisfies Pr[Z_P > t] ≤ 2e^-Ω(t) for all t > 0.