Optimal Vertex-Cut Sparsification of Quasi-Bipartite Graphs

TL;DR

This paper introduces a new vertex-cut sparsification method for quasi-bipartite graphs, reducing the size of the sparsifier to O(k^2) edges and vertices, improving previous bounds and providing matching lower bounds.

Contribution

It establishes a deterministic polynomial-time construction of sparsifiers with O(k^2) size for quasi-bipartite graphs, extending to graphs with small separators, and proves nearly tight lower bounds.

Findings

Achieved O(k^2) size sparsifiers for quasi-bipartite graphs.

Provided a deterministic polynomial-time construction method.

Proved nearly matching lower bounds for sparsifier size.

Abstract

In vertex-cut sparsification, given a graph $G=(V,E)$ with a terminal set $T\subseteq V$, we wish to construct a graph $G'=(V',E')$ with $T\subseteq V'$, such that for every two sets of terminals $A,B\subseteq T$, the size of a minimum $(A,B)$-vertex-cut in $G'$ is the same as in $G$. In the most basic setting, $G$ is unweighted and undirected, and we wish to bound the size of $G'$ by a function of $k=|T|$. Kratsch and Wahlstr\"om [JACM 2020] proved that every graph $G$ (possibly directed), admits a vertex-cut sparsifier $G'$ with $O(k^3)$ vertices, which can in fact be constructed in randomized polynomial time. We study (possibly directed) graphs $G$ that are quasi-bipartite, i.e., every edge has at least one endpoint in $T$, and prove that they admit a vertex-cut sparsifier with $O(k^2)$ edges and vertices, which can in fact be constructed in deterministic polynomial time. In fact, this bound naturally extends to all graphs with a small separator into bounded-size sets. Finally, we prove information-theoretically a nearly-matching lower bound, i.e., that $\tilde{\Omega}(k^2)$ edges are required to sparsify quasi-bipartite undirected graphs.