ABSTRACT
Suppose we are given a graph G = (V, E) and a set of terminals K ⊂ V. We consider the problem of constructing a graph H = (K, EH) that approximately preserves the congestion of every multicommodity flow with endpoints supported in K. We refer to such a graph as a flow sparsifier. We prove that there exist flow sparsifiers that simultaneously preserve the congestion of all multicommodity flows within an O(log k / log log k)-factor where |K| = k. This bound improves to O(1) if G excludes any fixed minor. This is a strengthening of previous results, which consider the problem of finding a graph H = (K, EH) (a cut sparsifier) that approximately preserves the value of minimum cuts separating any partition of the terminals. Indirectly our result also allows us to give a construction for better quality cut sparsifiers (and flow sparsifiers). Thereby, we immediately improve all approximation ratios derived using vertex sparsification in [14].
We also prove an Ω(log log k) lower bound for how well a flow sparsifier can simultaneously approximate the congestion of every multicommodity flow in the original graph. The proof of this theorem relies on a technique (which we refer to as oblivious dual certifcates) for proving super-constant congestion lower bounds against many multicommodity flows at once. Our result implies that approximation algorithms for multicommodity flow-type problems designed by a black box reduction to a "uniform" case on k nodes (see [14] for examples) must incur a super-constant cost in the approximation ratio.
- M. Adler, N. Harvey, K. Jain, R. Kleinberg, and A. R. Lehman. On the capacity of information networks. Symposium on Discrete Algorithms, pages 320--329, 1990.Google Scholar
- J. Batson, D. Spielman, and N. Srivastava. Twice-ramanujan sparsifiers. Symposium on Theory of Computing, pages 255--262, 2009. Google ScholarDigital Library
- A. Benczúr and D. Karger. Approximating s-t minimum cuts in O(n2) time. Symposium on Theory of Computing, pages 47--55, 1996. Google ScholarDigital Library
- A. Bhattacharyya, E. Grigorescu, K. M. Jung, S. Raskhodnikova, and D. Woodruff. Transitive-closure spanners. Symposium on Discrete Algorithms, pages 932--941, 2009. Google ScholarDigital Library
- D. Bienstock, E. Brickell, and C. Monma. On the structure of minimum-weight k-connected spanning networks. SIAM Journal on Discrete Mathematics, pages 320--329, 1990. Google ScholarDigital Library
- G. Calinescu, H. Karloff, and Y. Rabani. Approximation algorithms for the 0-extension problem. Symposium on Discrete Algorithms, pages 8--16, 2001. Google ScholarDigital Library
- Y. H. Chan, W. S. Fung, L. C. Lau, and C. K. Yung. Degree bounded network design with metric costs. Foundations of Computer Science, pages 125--134, 2008. Google ScholarDigital Library
- L. P. Chew. There is a planar graph almost as good as the complete graph. Symposium on Computational Geometry, pages 169--177, 1986. Google ScholarDigital Library
- J. Fakcharoenphol, C. Harrelson, S. Rao, and K. Talwar. An improved approximation algorithm for the 0-extension problem. Symposium on Discrete Algorithms, pages 257--265, 2003. Google ScholarDigital Library
- A. Gupta, V. Nagarajan, and R. Ravi. Improved approximation algorithm for requirement cut. submitted, 2009.Google Scholar
- N. Harvey, R. Kleinberg, and A. R. Lehman. On the capacity of information networks. IEEE Transactions on Information Theory, pages 251--260, 2006.Google Scholar
- A. Karzanov. Minimum 0-extensions of graph metrics. European Journal of Combinatorics, pages 71--101, 1998. Google ScholarDigital Library
- R. Khandekar, S. Rao, and U. Vazirani. Graph partitioning using single commodity flows. Journal of the ACM, 2009. Google ScholarDigital Library
- A. Moitra. Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. Foundations of Computer Science, pages 3--12, 2009. Google ScholarDigital Library
- D. Spielman and N. Srivastava. Graph sparsification by effective resistances. Symposium on Theory of Computing, pages 563--568, 2008. Google ScholarDigital Library
- D. Spielman and S. H. Teng. Nearly-linear time algorithms for graph partitioning, sparsification and solving linear systems. Symposium on Theory of Computing, pages 81--90, 2004. Google ScholarDigital Library
Index Terms
- Extensions and limits to vertex sparsification
Recommendations
Vertex Sparsifiers: New Results from Old Techniques
Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$...
Improved Guarantees for Vertex Sparsification in Planar Graphs
Graph sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study vertex sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on ...
Faster Approximation Algorithms for the Unit Capacity Concurrent Flow Problem with Applications to Routing and Finding Sparse Cuts
This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right,...
Comments