Towards (1+ε)-Approximate Flow Sparsifiers

Abstract

A useful approach to "compress" a large network $G$ is to represent it with a {\em flow-sparsifier}, i.e., a small network $H$ that supports the same flows as $G$, up to a factor $q\ge 1$ called the quality of sparsifier. Specifically, we assume the network $G$ contains a set of $k$ terminals $T$, shared with the network $H$, i.e., $T\subseteq V(G)\cap V(H)$, and we want $H$ to preserve all multicommodity flows that can be routed between the terminals $T$. The challenge is to construct $H$ that is small. These questions have received a lot of attention in recent years, leading to some known tradeoffs between the sparsifier's quality $q$ and its size $|V(H)|$. Nevertheless, it remains an outstanding question whether every $G$ admits a flow-sparsifier $H$ with quality $q=1+\epsilon$, or even $q=O(1)$, and size $|V(H)|\le f(k,\epsilon )$ (in particular, independent of $|V(G)|$ and the edge capacities). Making a first step in this direction, we present new constructions for several scenarios: * Our main result is that for quasi-bipartite networks $G$, one can construct a $(1+\epsilon )$-flow-sparsifier of size $\poly(k/\eps)$. In contrast, exact ($q=1$) sparsifiers for this family of networks are known to require size $2^{\Omega (k)}$. * For networks $G$ of bounded treewidth $w$, we construct a flow-sparsifier with quality $q=O(\log w/\log \log w)$ and size $O(w\cdot \poly(k))$. * For general networks $G$, we construct a {\em sketch} $sk(G)$, that stores all the feasible multicommodity flows up to factor $q=1+\eps$, and its size (storage requirement) is $f(k,\epsilon )$.

Full version of a paper accepted to SODA 2014

References in corpus (1)

• On Mimicking Networks Representing Minimum Terminal Cuts

Cited by in corpus (3)

• Refined Vertex Sparsifiers of Planar Graphs
• Vertex Sparsification for Edge Connectivity in Polynomial Time
• Mimicking Networks Parameterized by Connectivity