Cutting Corners Cheaply, or How to Remove Steiner Points

Abstract

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 $\beta=O(\log \lvert X\rvert)$, 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...