Vertex Sparsification in Trees

TL;DR

This paper introduces methods for creating high-quality vertex flow and cut sparsifiers in trees and quasi-bipartite graphs, establishing tight bounds and improved constructions for these graph classes.

Contribution

The paper presents the first tight bounds for vertex sparsifiers in trees and extends results to quasi-bipartite graphs with improved sparsifier constructions.

Findings

Achieved a 2-quality vertex flow and cut sparsifier for trees with size equal to terminals

Proved a tight lower bound of 2-o(1) for star graphs

Constructed exact sparsifiers of size O(2^k) for unweighted graphs

Abstract

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. In addition we give improved results for quasi-bipartite graphs. First, we show how to obtain a $2$-quality flow sparsifier with $V_H = K$ for such graphs. We then consider the other extreme and construct exact sparsifiers of size $O(2^{k})$, when the input graph is unweighted.