Near-linear-time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs

TL;DR

This paper presents a near-linear-time algorithm to compute vertex cut sparsifiers of size Θ(k^2) in directed acyclic graphs, improving previous bounds and ensuring the sparsifier preserves all min-cuts between terminal sets.

Contribution

It introduces a novel approach using total orderings inspired by extremal combinatorics to construct smaller sparsifiers for DAGs, improving the size and computation time over prior work.

Findings

Constructed vertex sparsifiers of size Θ(k^2) for DAGs.

Achieved near-linear time computation of the sparsifiers.

Proved that size Θ(k^2) is necessary for such sparsifiers.

Abstract

Let $G$ be a graph and $S, T \subseteq V(G)$ be (possibly overlapping) sets of terminals, $|S|=|T|=k$. We are interested in computing a vertex sparsifier for terminal cuts in $G$, i.e., a graph $H$ on a smallest possible number of vertices, where $S \cup T \subseteq V(H)$ and such that for every $A \subseteq S$ and $B \subseteq T$ the size of a minimum $(A,B)$-vertex cut is the same in $G$ as in $H$. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlstr\"om (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier $H$ with $O(k^3)$ vertices can be computed in randomized polynomial time, even for arbitrary digraphs $G$. However, since then, no improvements on the size $O(k^3)$ have been shown. In this paper, we draw inspiration from the renowned Bollob\'as's Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlstr\"om's methods. This new perspective allows us to construct a sparsifier $H$ of $\Theta(k^2)$ vertices for the case that $G$ is a DAG. We also show how to compute $H$ in time near-linear in the size of $G$, improving on the previous $O(n^{\omega+1})$. Furthermore, $H$ recovers the closest min-cut in $G$ for every partition $(A,B)$, which was not previously known. Finally, we show that a sparsifier of size $\Omega(k^2)$ is required, both for DAGs and for undirected edge cuts.