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Cops, robbers, and threatening skeletons: padded decomposition for minor-free graphs

Published: 31 May 2014 Publication History

Abstract

We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r/Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2/Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.

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References

[1]
Ittai Abraham, Cyril Gavoille, Dahlia Malkhi, and Udi Wieder. Strong-diameter decompositions of minor free graphs. Theory Comput. Syst., 47(4):837--855, 2010.
[2]
Noga Alon, Paul Seymour, and Robin Thomas. A separator theorem for nonplanar graphs. J. Amer. Math. Soc., 3(4):801--808, 1990.
[3]
Thomas Andreae. On a pursuit game played on graphs for which a minor is excluded. J. Combin. Theory Ser. B, 41(1):37--47, 1986.
[4]
Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1079--1087, New York, 2004. ACM.
[5]
Baruch Awerbuch. Complexity of network synchronization. J. ACM, 32(4):804--823, October 1985.
[6]
Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS '96, pages 184--, Washington, DC, USA, 1996. IEEE Computer Society.
[7]
Punyashloka Biswal, James R. Lee, and Satish Rao. Eigenvalue bounds, spectral partitioning, and metrical deformations via ows. J. ACM, 57(3), 2010.
[8]
Glencora Borradaile, James R. Lee, and Anastasios Sidiropoulos. Randomly removing g handles at once. Comput. Geom., 43(8):655--662, 2010.
[9]
Costas Busch, Ryan LaFortune, and Srikanta Tirthapura. Improved sparse covers for graphs excluding a fixed minor. In Proceedings of the twenty-sixth annual ACM symposium on Principles of distributed computing, PODC '07, pages 61--70, New York, NY, USA, 2007. ACM.
[10]
Gruia Calinescu, Howard Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. SIAM J. Comput., 34(2):358--372, 2004/05.
[11]
Reinhard Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2000.
[12]
Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. System Sci., 69(3):485--497, 2004.
[13]
Jittat Fakcharoenphol and Kunal Talwar. An improved decomposition theorem for graphs excluding a fixed minor. RANDOM-APPROX, pages 36--46, 2003.
[14]
Uriel Feige, MohammadTaghi Hajiaghayi, and James R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput., 38(2):629--657, 2008.
[15]
Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low--distortion embeddings. In FOCS, pages 534--543, 2003.
[16]
Piotr Indyk and Anastasios Sidiropoulos. Probabilistic embeddings of bounded genus graphs into planar graphs. In Symposium on Computational Geometry, pages 204--209, 2007.
[17]
Jonathan A. Kelner, James R. Lee, Gregory N. Price, and Shang-Hua Teng. Higher eigenvalues of graphs. In FOCS, pages 735--744, 2009.
[18]
Philip N. Klein, Serge A. Plotkin, and Satish Rao. Excluded minors, network decomposition, and multicommodity flow. In STOC, pages 682--690, 1993.
[19]
James R. Lee. Open question recap, February 2013. http://tcsmath.wordpress.com/2013/02/25/openquestion-recap/.
[20]
James R. Lee, Shayan Oveis Gharan, and Luca Trevisan. Multi-way spectral partitioning and higher-order cheeger inequalities. In STOC, pages 1117--1130, 2012.
[21]
James R. Lee and Assaf Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1):59--95, 2005.
[22]
James R. Lee and Anastasios Sidiropoulos. Genus and the geometry of the cut graph. In SODA, pages 193--201, 2010.
[23]
Nathan Linial and Michael Saks. Low diameter graph decompositions. Combinatorica, 13(4):441--454, 1993. (Preliminary version in 2nd SODA, 1991).
[24]
Jiří Matoušek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002.
[25]
Serge Plotkin, Satish Rao, and Warren D. Smith. Shallow excluded minors and improved graph decompositions. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), pages 462--470, New York, 1994. ACM.
[26]
Yuri Rabinovich. On average distortion of embedding metrics into the line and into ℓ1. In Proceedings of the thirty-fifth ACM symposium on Theory of computing, pages 456--462. ACM Press, 2003.
[27]
Satish B. Rao. Small distortion and volume preserving embeddings for planar and Euclidean metrics. In SOCG, pages 300--306, 1999.
[28]
Neil Robertson and Paul D. Seymour. Graph minors. XVI. excluding a non-planar graph. Journal of Combinatorial Theory, Series B, 89(1):43--76, 2003.
[29]
Anastasios Sidiropoulos. Optimal stochastic planarization. In FOCS, pages 163--170, 2010.
[30]
Christian Wulff-Nilsen. Separator theorems for minor-free and shallow minor-free graphs with applications. In FOCS, pages 37--46, 2011.

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    cover image ACM Conferences
    STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing
    May 2014
    984 pages
    ISBN:9781450327107
    DOI:10.1145/2591796
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    Published: 31 May 2014

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    Author Tags

    1. cops and robbers
    2. excluded minor
    3. padded decomposition

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    May 31 - June 3, 2014
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    • (2023)Approximate Max-Flow Min-Multicut Theorem for Graphs of Bounded TreewidthProceedings of the 55th Annual ACM Symposium on Theory of Computing10.1145/3564246.3585150(1325-1334)Online publication date: 2-Jun-2023
    • (2021)Conformal growth rates and spectral geometry on distributional limits of graphsThe Annals of Probability10.1214/20-AOP148049:6Online publication date: 1-Nov-2021
    • (2020)A tight lower bound for the capture time of the Cops and Robbers gameTheoretical Computer Science10.1016/j.tcs.2020.06.004Online publication date: Jun-2020
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    • (2017)Metric Decompositions of Path-Separable GraphsAlgorithmica10.1007/s00453-016-0213-079:3(645-653)Online publication date: 1-Nov-2017
    • (2016)Cheeger-Type Approximation for Sparsest st-CutACM Transactions on Algorithms10.1145/299679913:1(1-21)Online publication date: 19-Nov-2016

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