Towards (1 + ε)-Approximate flow sparsifiers

A useful approach to "compress" a large network G is to represent it with a flow-sparsifier, 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 flow-sparsifier H with quality q = 1 + ε, or even q = O(1), and size |V(H)| < f(k, ε) (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 + ε)-flow-sparsifier of size poly(k/ ε). 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 flow-sparsifier 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 + ε, and its size (storage requirement) is f(k, ε).