NDLI: Vertex Sparsifiers and Abstract Rounding Algorithms

Content Provider	IEEE Xplore Digital Library
Author	Charikar, M. Leighton, T. Shi Li Moitra, A.
Copyright Year	2010
Abstract	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.
Starting Page	265
Ending Page	274
File Size	402414
Page Count	10
File Format	PDF
ISBN	9781424485253
ISSN	02725428
DOI	10.1109/FOCS.2010.32
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2010-10-23
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Approximation methods Approximation algorithms Routing Polynomials Measurement Optimization Hamming weight approximation algorithms vertex sparsifier
Content Type	Text
Resource Type	Article

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