Vertex Sparsifiers and Abstract Rounding Algorithms

The notion of vertex sparsification (in particular cut-sparsification) is introduced in (Moitra, 2009), where it was shown that for any graph $G = (V, E)$ and any subset of $k$ terminals $K \subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ \emph{on just the terminal set} so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. Then approximation algorithms can be run directly on $H$ as a proxy for running on $G$. We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. %In fact, we prove that in general we cannot hope for approximation factors better than We prove a lower bound of $\Omega(\log^{1/4} k)$ – this is polynomially-related to the known upper bound of $O(\log k/\log \log k)$. Independently, a similar lower bound is given in (Makarychev, Makarychev, 2010). This is an exponential improvement on the $\Omega(\log \log k)$ bound given in (Leighton, Moitra, 2010) which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal $O(\log k)$-competitive Steiner oblivious routing schemes, which generalize the results in (Raecke, 2008). We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(\log k)$ times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earth-mover constraints.