Exact Flow Sparsification Requires Unbounded Size

Abstract: Given a large edge-capacitated network $G$ and a subset of $k$ vertices called terminals, an (exact) flow sparsifier is a small network $G'$ that preserves (exactly) all multicommodity flows that can be routed between the terminals. Flow sparsifiers were introduced by Leighton and Moitra [STOC 2010], and have been studied and used in many algorithmic contexts. A fundamental question that remained open for over a decade, asks whether every $k$-terminal network admits an exact flow sparsifier whose size is bounded by some function $f(k)$ (regardless of the size of $G$ or its capacities). We resolve this question in the negative by proving that there exist $6$-terminal networks $G$ whose flow sparsifiers $G'$ must have arbitrarily large size. This unboundedness is perhaps surprising, since the analogous sparsification that preserves all terminal cuts (called exact cut sparsifier or mimicking network) admits sparsifiers of size $f_0(k)\leq 2^{2^k}$ [Hagerup, Katajainen, Nishimura, and Ragde, JCSS 1998]. We prove our results by analyzing the set of all feasible demands in the network, known as the demand polytope. We identify an invariant of this polytope, essentially the slope of certain facets, that can be made arbitrarily large even for $k=6$, and implies an explicit lower bound on the size of the network. We further use this technique to answer, again in the negative, an open question of Seymour [JCTB 2015] regarding flow-sparsification that uses only contractions and preserves the infeasibility of one demand vector.