On $(1+\varepsilon)$-Approximate Flow Sparsifiers

TL;DR

This paper investigates the construction of small, high-quality flow sparsifiers for graphs with terminal sets, establishing existence results for 5 terminals and hardness for 6, using metric geometry tools.

Contribution

It proves the existence of $(1+ ext{ε})$-quality contraction-based flow sparsifiers for 5-terminal graphs and shows non-existence for 6-terminals, advancing understanding of graph compression limits.

Findings

Existence of $(1+ε)$-quality flow sparsifiers for 5-terminal graphs.

Non-existence of such sparsifiers for 6-terminal graphs.

Utilization of tight spans in metric geometry for the proofs.

Abstract

Given a large graph $G$ with a subset $|T|=k$ of its vertices called terminals, a quality-$q$ flow sparsifier is a small graph $G'$ that contains $T$ and preserves all multicommodity flows that can be routed between terminals in $T$, to within factor $q$. The problem of constructing flow sparsifiers with good (small) quality and (small) size has been a central problem in graph compression for decades. A natural approach of constructing $O(1)$-quality flow sparsifiers, which was adopted in most previous constructions, is contraction. Andoni, Krauthgamer, and Gupta constructed a sketch of size $f(k,\varepsilon)$ that stores all feasible multicommodity flows up to a factor of $(1+\varepsilon)$, raised the question of constructing quality-$(1+\varepsilon)$ flow sparsifiers whose size only depends on $k,\varepsilon$ (but not the number of vertices in the input graph $G$), and proposed a contraction-based framework towards it using their sketch result. In this paper, we settle their question for contraction-based flow sparsifiers, by showing that quality-$(1+\varepsilon)$ contraction-based flow sparsifiers with size $f(\varepsilon)$ exist for all $5$-terminal graphs, but not for all $6$-terminal graphs. Our hardness result on $6$-terminal graphs improves upon a recent hardness result by Krauthgamer and Mosenzon on exact (quality-$1$) flow sparsifiers, for contraction-based constructions. Our construction and proof utilize the notion of tight spans in metric geometry, which we believe is a powerful tool for future work.