O(1) Steiner Point Removal in Series-Parallel Graphs
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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' ⊆ 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.
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