On strong diameter padded decompositions

Arnold Filtser

doi:10.4230/LIPIcs.APPROX-RANDOM.2019.6

On strong diameter padded decompositions

Arnold Filtser

Ben-Gurion University of the Negev

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

16 Scopus citations

Abstract

Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least e⁻β·^γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r²) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), Õ(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

Original language	English
Title of host publication	Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
Editors	Dimitris Achlioptas, Laszlo A. Vegh
Publisher	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)	9783959771252
DOIs	https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.6
State	Published - Sep 2019
Externally published	Yes
Event	22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States Duration: 20 Sep 2019 → 22 Sep 2019

Publication series

Name	Leibniz International Proceedings in Informatics, LIPIcs
Volume	145
ISSN (Print)	1868-8969

Conference

Conference	22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
Country/Territory	United States
City	Cambridge
Period	20/09/19 → 22/09/19

Bibliographical note

Publisher Copyright:
© Arnold Filtser.

Funding

Arnold Filtser: Supported in part by ISF grant No. (1817/17) and by BSF grant No. 2015813. The author would like to thank Ofer Neiman for helpful discussions. in part by ISF grant No. (1817/17) and by BSF grant No.

Funders	Funder number
United States-Israel Binational Science Foundation	2015813
Israel Science Foundation	1817/17

Keywords

Distance oracles
Doubling dimension
Minor free graphs
Padded decomposition
Spanners
Sparse cover
Strong diameter
Unique games

Access to Document

10.4230/LIPIcs.APPROX-RANDOM.2019.6

Cite this

Filtser, A. (2019). On strong diameter padded decompositions. In D. Achlioptas, & L. A. Vegh (Eds.), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019 Article 6 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 145). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.6

@inproceedings{9a911c2bc8d1494089a2e51c4d4a5450,

title = "On strong diameter padded decompositions",

abstract = "Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least e−β·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), {\~O}(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.",

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Filtser, A 2019, On strong diameter padded decompositions. in D Achlioptas & LA Vegh (eds), Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019., 6, Leibniz International Proceedings in Informatics, LIPIcs, vol. 145, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019, Cambridge, United States, 20/09/19. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.6

On strong diameter padded decompositions. / Filtser, Arnold.
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019. ed. / Dimitris Achlioptas; Laszlo A. Vegh. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 6 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 145).

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

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N2 - Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least e−β·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), Õ(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.

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KW - Strong diameter

KW - Unique games

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Filtser A. On strong diameter padded decompositions. In Achlioptas D, Vegh LA, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2019. 6. (Leibniz International Proceedings in Informatics, LIPIcs). doi: 10.4230/LIPIcs.APPROX-RANDOM.2019.6

On strong diameter padded decompositions

Abstract

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Bibliographical note

Funding

Keywords

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