On Vertex Sparsifiers with Steiner Nodes

TL;DR

This paper explores the construction of vertex sparsifiers with Steiner nodes that approximate cut and flow properties in graphs, providing algorithms for constant-quality sparsifiers with size bounds depending on terminal capacities.

Contribution

It introduces algorithms for creating constant-quality cut and flow sparsifiers that include Steiner nodes, extending previous work limited to terminal-only sparsifiers.

Findings

Constructed constant-quality cut sparsifiers of size $O(C^3)$

Developed flow sparsifiers of size $C^{O(\log\log C)}$

Provided algorithms with polynomial and exponential time complexities

Abstract

Given an undirected graph $G=(V,E)$ with edge capacities $c_e\geq 1$ for $e\in E$ and a subset $T$ of $k$ vertices called terminals, we say that a graph $H$ is a quality-$q$ cut sparsifier for $G$ iff $T\subseteq V(H)$, and for any partition $(A,B)$ of $T$, the values of the minimum cuts separating $A$ and $B$ in graphs $G$ and $H$ are within a factor $q$ from each other. We say that $H$ is a quality-$q$ flow sparsifier for $G$ iff $T\subseteq V(H)$, and for any set $D$ of demands over the terminals, the values of the minimum edge congestion incurred by fractionally routing the demands in $D$ in graphs $G$ and $H$ are within a factor $q$ from each other. So far vertex sparsifiers have been studied in a restricted setting where the sparsifier $H$ is not allowed to contain any non-terminal vertices, that is $V(H)=T$. For this setting, efficient algorithms are known for constructing quality-$O(\log k/\log\log k)$ cut and flow vertex sparsifiers, as well as a lower bound of $\tilde{\Omega}(\sqrt{\log k})$ on the quality of any flow or cut sparsifier. We study flow and cut sparsifiers in the more general setting where Steiner vertices are allowed, that is, we no longer require that $V(H)=T$. We show algorithms to construct constant-quality cut sparsifiers of size $O(C^3)$ in time $\poly(n)\cdot 2^C$, and constant-quality flow sparsifiers of size $C^{O(\log\log C)}$ in time $n^{O(\log C)}\cdot 2^C$, where $C$ is the total capacity of the edges incident on the terminals.