$O(1)$ Steiner Point Removal in Series-Parallel Graphs

TL;DR

This paper presents an $O(1)$ distortion Steiner point removal solution for series-parallel graphs, extending previous results from simpler graph classes by introducing a novel hammock decomposition technique.

Contribution

It introduces a new metric decomposition called hammock decomposition that enables $O(1)$ SPR solutions for series-parallel graphs, expanding the class of graphs with such solutions.

Findings

Achieved $O(1)$ distortion SPR for series-parallel graphs.

Developed a new hammock decomposition technique.

Extended SPR results to a broader class of graphs.

Abstract

We study how to vertex-sparsify a graph while preserving both the graph's metric and structure. Specifically, we study the Steiner point removal (SPR) problem where we are given a weighted graph $G=(V,E,w)$ and terminal set $V' \subseteq V$ and must compute a weighted minor $G'=(V',E', w')$ of $G$ which approximates $G$'s metric on $V'$. A major open question in the area of metric embeddings is the existence of $O(1)$ multiplicative distortion SPR solutions for every (non-trivial) minor-closed family of graphs. To this end prior work has studied SPR on trees, cactus and outerplanar graphs and showed that in these graphs such a minor exists with $O(1)$ distortion. We give $O(1)$ distortion SPR solutions for series-parallel graphs, extending the frontier of this line of work. The main engine of our approach is a new metric decomposition for series-parallel graphs which we call a hammock decomposition. Roughly, a hammock decomposition is a forest-like structure that preserves certain critical parts of the metric induced by a series-parallel graph.