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Approximating the Norms of Graph Spanners

Authors Eden Chlamtáč, Michael Dinitz, Thomas Robinson

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Author Details

Eden Chlamtáč
  • Ben Gurion University of the Negev, Beersheva, Israel
Michael Dinitz
  • Johns Hopkins University, Baltimore, MD, USA
Thomas Robinson
  • Ben Gurion University of the Negev, Beersheva, Israel

Funding

  • Chlamtáč, Eden: Supported in part by ISF grant 1002/14.
  • Dinitz, Michael: Supported in part by NSF awards CCF-1464239 and CCF-1535887.
  • Robinson, Thomas: Supported in part by ISF grant 1002/14.

Cite AsGet BibTex

Eden Chlamtáč, Michael Dinitz, and Thomas Robinson. Approximating the Norms of Graph Spanners. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 11:1-11:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.11

Abstract

The l_p-norm of the degree vector was recently introduced by [Chlamtáč, Dinitz, Robinson ICALP '19] as a new cost metric for graph spanners, as it interpolates between two traditional notions of cost (the sparsity l_1 and the max degree l_infty) and is well-motivated from applications. We study this from an approximation algorithms point of view, analyzing old algorithms and designing new algorithms for this new context, as well as providing hardness results. Our main results are for the l_2-norm and stretch 3, where we give a tight analysis of the greedy algorithm and a new algorithm specifically tailored to this setting which gives an improved approximation ratio.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
Keywords
  • Spanners
  • Approximations

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References

  1. Noga Alon, Yossi Azar, Gerhard J. Woeginger, and Tal Yadid. Approximation Schemes for Scheduling. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '97, 1997. Google Scholar
  2. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete Comput. Geom., 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof Verification and the Hardness of Approximation Problems. J. ACM, 45(3):501-555, May 1998. URL: https://doi.org/10.1145/278298.278306.
  4. Sanjeev Arora and Shmuel Safra. Probabilistic Checking of Proofs: A New Characterization of NP. J. ACM, 45(1):70-122, January 1998. URL: https://doi.org/10.1145/273865.273901.
  5. Yossi Azar, Leah Epstein, Yossi Richter, and Gerhard J. Woeginger. All-Norm Approximation Algorithms. In Martti Penttonen and Erik Meineche Schmidt, editors, Algorithm Theory - SWAT 2002, 2002. Google Scholar
  6. Nikhil Bansal and Kirk Pruhs. Server Scheduling in the L_p Norm: A Rising Tide Lifts All Boat. In Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, STOC '03, pages 242-250, 2003. Google Scholar
  7. Piotr Berman, Arnab Bhattacharyya, Konstantin Makarychev, Sofya Raskhodnikova, and Grigory Yaroslavtsev. Approximation algorithms for spanner problems and Directed Steiner Forest. Inf. Comput., 222:93-107, 2013. URL: https://doi.org/10.1016/j.ic.2012.10.007.
  8. Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. New Sparseness Results on Graph Spanners. In Proceedings of the Eighth Annual Symposium on Computational Geometry, SCG '92, pages 192-201, New York, NY, USA, 1992. ACM. URL: https://doi.org/10.1145/142675.142717.
  9. Eden Chlamtáč and Michael Dinitz. Lowest-Degree k-Spanner: Approximation and Hardness. Theory of Computing, 12(15):1-29, 2016. URL: https://doi.org/10.4086/toc.2016.v012a015.
  10. Eden Chlamtáč, Michael Dinitz, and Thomas Robinson. The Norms of Graph Spanners. In Proceedings of the 46th International Colloquium Conference on Automata, Languages, and Programming, ICALP '19, 2019. Google Scholar
  11. Michael Dinitz, Guy Kortsarz, and Ran Raz. Label Cover Instances with Large Girth and the Hardness of Approximating Basic k-Spanner. ACM Trans. Algorithms, 12(2):25:1-25:16, December 2015. URL: https://doi.org/10.1145/2818375.
  12. Michael Dinitz and Robert Krauthgamer. Directed Spanners via Flow-based Linear Programs. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC '11, pages 323-332, New York, NY, USA, 2011. ACM. URL: https://doi.org/10.1145/1993636.1993680.
  13. Michael Dinitz and Zeyu Zhang. Approximating Low-stretch Spanners. In Proceedings of the Twenty-seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '16, pages 821-840, Philadelphia, PA, USA, 2016. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2884435.2884494.
  14. Devdatt P. Dubhashi and Alessandro Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009. URL: http://www.cambridge.org/gb/knowledge/isbn/item2327542/.
  15. Michael Elkin and David Peleg. The Hardness of Approximating Spanner Problems. Theor. Comp. Sys., 41(4):691-729, December 2007. URL: https://doi.org/10.1007/s00224-006-1266-2.
  16. Daniel Golovin, Anupam Gupta, Amit Kumar, and Kanat Tangwongsan. All-Norms and All-L_p-Norms Approximation Algorithms. In FSTTCS, volume 2 of LIPIcs, pages 199-210. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2008. Google Scholar
  17. Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Computational Geometry, 28(2):89-112, 2004. Special Issue on the 18th Annual Symposium on Computational Geometry - SoCG2002. URL: https://doi.org/10.1016/j.comgeo.2004.03.003.
  18. Guy Kortsarz. On the Hardness of Approximating Spanners. Algorithmica, 30(3):432-450, 2001. URL: https://doi.org/10.1007/s00453-001-0021-y.
  19. S. Lloyd. Least squares quantization in PCM. IEEE Transactions on Information Theory, 28(2):129-137, March 1982. URL: https://doi.org/10.1109/TIT.1982.1056489.
  20. Ran Raz. A Parallel Repetition Theorem. SIAM J. Comput., 27(3):763-803, 1998. URL: https://doi.org/10.1137/S0097539795280895.
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