Vertex sparsifiers: New results from old techniques

Matthias Englert; Anupam Gupta; Robert Krauthgamer; Harald Räcke; Inbal Talgam-Cohen; Kunal Talwar

doi:10.1007/978-3-642-15369-3_12

Vertex sparsifiers: New results from old techniques

Matthias Englert^*, Anupam Gupta, Robert Krauthgamer, Harald Räcke, Inbal Talgam-Cohen, Kunal Talwar

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

20 Scopus citations

Abstract

Given a capacitated graph G = (V,E) and a set of terminals K ⊆ V, how should we produce a graph H only on the terminals K so that every (multicommodity) flow between the terminals in G could be supported in H with low congestion, and vice versa? (Such a graph H is called a flow-sparsifier for G.) What if we want H to be a "simple" graph? What if we allow H to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier H that maintains congestion up to a factor of O(log k)/log log k), where k = |K|. (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(logk). (c) for a planar graph G, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in G. Moreover, this result extends to minor-closed families of graphs. Our bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization
Subtitle of host publication	Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings
Pages	152-165
Number of pages	14
DOIs	https://doi.org/10.1007/978-3-642-15369-3_12
State	Published - 2010
Externally published	Yes
Event	13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010 - Barcelona, Spain Duration: 1 Sep 2010 → 3 Sep 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6302 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010
Country/Territory	Spain
City	Barcelona
Period	1/09/10 → 3/09/10

Funding

Funders	Funder number
Directorate for Computer and Information Science and Engineering	0729022
Engineering and Physical Sciences Research Council	EP/F043333/1

Access to Document

10.1007/978-3-642-15369-3_12

Cite this

Englert, M., Gupta, A., Krauthgamer, R., Räcke, H., Talgam-Cohen, I., & Talwar, K. (2010). Vertex sparsifiers: New results from old techniques. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings (pp. 152-165). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6302 LNCS). https://doi.org/10.1007/978-3-642-15369-3_12

Englert, Matthias ; Gupta, Anupam ; Krauthgamer, Robert et al. / Vertex sparsifiers : New results from old techniques. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. 2010. pp. 152-165 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Vertex sparsifiers: New results from old techniques",

abstract = "Given a capacitated graph G = (V,E) and a set of terminals K ⊆ V, how should we produce a graph H only on the terminals K so that every (multicommodity) flow between the terminals in G could be supported in H with low congestion, and vice versa? (Such a graph H is called a flow-sparsifier for G.) What if we want H to be a {"}simple{"} graph? What if we allow H to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier H that maintains congestion up to a factor of O(log k)/log log k), where k = |K|. (b) a convex combination of trees over the terminals K that maintains congestion up to a factor of O(logk). (c) for a planar graph G, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in G. Moreover, this result extends to minor-closed families of graphs. Our bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.",

author = "Matthias Englert and Anupam Gupta and Robert Krauthgamer and Harald R{\"a}cke and Inbal Talgam-Cohen and Kunal Talwar",

year = "2010",

doi = "10.1007/978-3-642-15369-3_12",

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Englert, M, Gupta, A, Krauthgamer, R, Räcke, H, Talgam-Cohen, I & Talwar, K 2010, Vertex sparsifiers: New results from old techniques. in Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6302 LNCS, pp. 152-165, 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010, Barcelona, Spain, 1/09/10. https://doi.org/10.1007/978-3-642-15369-3_12

Vertex sparsifiers: New results from old techniques. / Englert, Matthias; Gupta, Anupam; Krauthgamer, Robert et al.
Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. 2010. p. 152-165 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6302 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Räcke, Harald

AU - Talgam-Cohen, Inbal

AU - Talwar, Kunal

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Englert M, Gupta A, Krauthgamer R, Räcke H, Talgam-Cohen I, Talwar K. Vertex sparsifiers: New results from old techniques. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings. 2010. p. 152-165. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-15369-3_12

Vertex sparsifiers: New results from old techniques

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