Date of Original Version
12014
Type
Conference Proceeding
Rights Management
Copyright © 2014 by the Society for Industrial and Applied Mathematics
Abstract or Description
A useful approach to “compress” a large network G is to represent it with a flowsparsifier, i.e., a small network H that supports the same flows as G, up to a factor q ≥ 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 ⊆ V(G)∩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 flowsparsifier H with quality q = 1 + ∊, or even q = O(1), and size V(H) ≤ f(k, ∊) (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 quasibipartite networks G, one can construct a (1 + ∊)flowsparsifier of size poly(k/∊). In contrast, exact (q = 1) sparsifiers for this family of networks are known to require size 2^{Ω(k)}.

For networks G of bounded treewidth w, we construct a flowsparsifier with quality q = O(logw/loglogw) and size O(w·poly(k)).

For general networks G, we construct a sketch sk(G), that stores all the feasible multicommodity flows up to factor q = 1 + ∊, and its size (storage requirement) is f(k, ∊).
DOI
10.1137/1.9781611973402.20
Published In
Proceedings of the ACMSIAM Symposium on Discrete Algorithms (SODA), 2014, 279293.