One Tree to Rule Them All: Poly-Logarithmic Universal Steiner Tree

TL;DR

This paper presents polynomial-time algorithms for constructing poly-logarithmic universal Steiner trees and strong sparse partition hierarchies, resolving longstanding open questions and improving results for special graph classes.

Contribution

It provides the first polynomial-time algorithms for poly-logarithmic USTs and strong sparse partition hierarchies, and tightens bounds for graphs with constant doubling dimension or pathwidth.

Findings

Achieved $O( ext{log}^7 n)$-approximate USTs for general graphs.

Established $O( ext{log} n)$-approximate USTs for graphs with constant doubling dimension or pathwidth.

Connected the existence of these structures to cluster aggregation and dangling nets problems.

Abstract

A spanning tree $T$ of graph $G$ is a $\rho$-approximate universal Steiner tree (UST) for root vertex $r$ if, for any subset of vertices $S$ containing $r$, the cost of the minimal subgraph of $T$ connecting $S$ is within a $\rho$ factor of the minimum cost tree connecting $S$ in $G$. Busch et al. (FOCS 2012) showed that every graph admits $2^{O(\sqrt{\log n})}$-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions. We settle these open questions by giving polynomial-time algorithms for computing both $O(\log ^ 7 n)$-approximate USTs and poly-logarithmic strong sparse partition hierarchies. For graphs with constant doubling dimension or constant pathwidth we improve this to $O(\log n)$-approximate USTs and $O(1)$ strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms. We reduce the existence of these objects to the previously studied cluster aggregation problem and what we call dangling nets.