Lower Bounds on Flow Sparsifiers with Steiner Nodes

TL;DR

This paper establishes lower bounds on the quality of flow sparsifiers with Steiner nodes, showing that allowing many Steiner nodes does not significantly improve the quality of contraction-based sparsifiers.

Contribution

It proves that even with a large number of Steiner nodes, the quality of flow sparsifiers cannot be substantially improved, providing fundamental lower bounds.

Findings

Flow sparsifiers with Steiner nodes still have poor quality bounds.

Contraction-based methods cannot significantly improve flow preservation.

Lower bounds hold even with exponentially many Steiner nodes.

Abstract

Given a large graph $G$ with a set of its $k$ vertices called terminals, a \emph{quality-$q$ flow sparsifier} is a small graph $G'$ that contains the terminals and preserves all multicommodity flows between them up to some multiplicative factor $q\ge 1$, called the \emph{quality}. Constructing flow sparsifiers with good quality and small size ($|V(G')|$) has been a central problem in graph compression. The most common approach of constructing flow sparsifiers is contraction: first compute a partition of the vertices in $V(G)$, and then contract each part into a supernode to obtain $G'$. When $G'$ is only allowed to contain all terminals, the best quality is shown to be $O(\log k/\log\log k)$ and $\Omega(\sqrt{\log k/\log\log k})$. In this paper, we show that allowing a few Steiner nodes does not help much in improving the quality. Specifically, there exist $k$-terminal graphs such that, even if we allow $k\cdot 2^{(\log k)^{\Omega(1)}}$ Steiner nodes in its contraction-based flow sparsifier, the quality is still $\Omega\big((\log k)^{0.3}\big)$.