Cutting corners cheaply, or how to remove Steiner points.

*(English)*Zbl 06473198Summary: 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 $T \subseteq V$, there is a graph $G'=(T,E',w')$ that is isomorphic to a minor of $G$ such that for every two terminals $u,v\in T$, the shortest-path distances between them in $G$ and in $G'$ satisfy $d_{G,w}(u,v) \le d_{G',w'}(u,v) \le O(\log^5|T|) \cdot d_{G,w}(u,v)$. 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 $\beta$-separating decomposition for $\beta=O(\log \vert X\vert)$, which means that for every $\Delta>0$ there is a randomized partitioning of $X$ into clusters of diameter at most $\Delta$, satisfying the following separation property: for every $x,y \in X$, the probability that they lie in different clusters of the partition is at most $\beta\,d(x,y)/\Delta$. We introduce an additional requirement in the form of a tail bound: for every shortest-path $P$ of length $d(P) \leq \Delta/\beta$, the number of clusters of the partition that meet the path $P$, denoted by $Z_P$, satisfies $\Pr[Z_P > t] \le 2e^{-\Omega(t)}$ for all $t>0$.