Vertex Sparsification in Trees

Given an unweighted tree \(T=(V,E)\) with terminals \(K \subset V\), we show how to obtain a 2-quality vertex flow and cut sparsifier H with \(V_H = K\). We prove that our result is essentially tight by providing a \(2-o(1)\) lower-bound on the quality of any cut sparsifier for stars.

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