Towards (1 + ∊)-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 + ∊, 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 quasi-bipartite networks G, one can construct a (1 + ∊)-flow-sparsifier 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 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 + ∊, and its size (storage requirement) is f(k, ∊).