Document Open Access Logo

Probabilistic Metric Embedding via Metric Labeling

Authors Kamesh Munagala, Govind S. Sankar, Erin Taylor



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2023.2.pdf
  • Filesize: 0.6 MB
  • 10 pages

Document Identifiers

Author Details

Kamesh Munagala
  • Department of Computer Science, Duke University, Durham, NC, USA
Govind S. Sankar
  • Department of Computer Science, Duke University, Durham, NC, USA
Erin Taylor
  • Department of Computer Science, Duke University, Durham, NC, USA

Cite AsGet BibTex

Kamesh Munagala, Govind S. Sankar, and Erin Taylor. Probabilistic Metric Embedding via Metric Labeling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 2:1-2:10, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.2

Abstract

We consider probabilistic embedding of metric spaces into ultra-metrics (or equivalently to a constant factor, into hierarchically separated trees) to minimize the expected distortion of any pairwise distance. Such embeddings have been widely used in network design and online algorithms. Our main result is a polynomial time algorithm that approximates the optimal distortion on any instance to within a constant factor. We achieve this via a novel LP formulation that reduces this problem to a probabilistic version of uniform metric labeling.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random projections and metric embeddings
Keywords
  • Metric Embedding
  • Approximation Algorithms
  • Ultrametrics

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Noga Alon, Mihai Bădoiu, Erik D. Demaine, Martin Farach-Colton, Mohammadtaghi Hajiaghayi, and Anastasios Sidiropoulos. Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics. ACM Trans. Algorithms, 4(4), August 2008. Google Scholar
  2. Noga Alon, Richard M. Karp, David Peleg, and Douglas West. A graph-theoretic game and its application to the k-server problem. SIAM Journal on Computing, 24(1):78-100, 1995. Google Scholar
  3. B. Awerbuch and Y. Azar. Buy-at-bulk network design. In Proceedings 38th Annual Symposium on Foundations of Computer Science, pages 542-547, 1997. URL: https://doi.org/10.1109/SFCS.1997.646143.
  4. Nikhil Bansal, Niv Buchbinder, Aleksander Madry, and Joseph (Seffi) Naor. A polylogarithmic-competitive algorithm for the k-server problem. J. ACM, 62(5), November 2015. Google Scholar
  5. Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of 37th Conference on Foundations of Computer Science, pages 184-193, 1996. URL: https://doi.org/10.1109/SFCS.1996.548477.
  6. Yair Bartal. On approximating arbitrary metrices by tree metrics. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC '98, pages 161-168, New York, NY, USA, 1998. Association for Computing Machinery. URL: https://doi.org/10.1145/276698.276725.
  7. Yair Bartal, Nathan Linial, Manor Mendel, and Assaf Naor. On metric Ramsey-type phenomena. Annals of Mathematics, 162(2):643-709, 2005. Google Scholar
  8. Sayan Bhattacharya, Gagan Goel, Sreenivas Gollapudi, and Kamesh Munagala. Budget constrained auctions with heterogeneous items. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 379-388. ACM, 2010. Google Scholar
  9. Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. Optimal multi-dimensional mechanism design: Reducing revenue to welfare maximization. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 130-139. IEEE Computer Society, 2012. Google Scholar
  10. Gunnar Carlsson and Facundo Mémoli. Characterization, stability and convergence of hierarchical clustering methods. Journal of Machine Learning Research, 11(47):1425-1470, 2010. URL: http://jmlr.org/papers/v11/carlsson10a.html.
  11. Moses Charikar, Chandra Chekuri, Ashish Goel, Sudipto Guha, and Serge Plotkin. Approximating a finite metric by a small number of tree metrics. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science, FOCS '98, page 379, USA, 1998. IEEE Computer Society. Google Scholar
  12. Shuchi Chawla, Jason D. Hartline, David L. Malec, and Balasubramanian Sivan. Multi-parameter mechanism design and sequential posted pricing. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 311-320. ACM, 2010. Google Scholar
  13. Chandra Chekuri, Anupam Gupta, Ilan Newman, Yuri Rabinovich, and Alistair Sinclair. Embedding k-outerplanar graphs into l1. SIAM Journal on Discrete Mathematics, 20(1):119-136, 2006. URL: https://doi.org/10.1137/S0895480102417379.
  14. Ning Chen, Nicole Immorlica, Anna R Karlin, Mohammad Mahdian, and Atri Rudra. Approximating matches made in heaven. In International Colloquium on Automata, Languages, and Programming, pages 266-278. Springer, 2009. Google Scholar
  15. Vincent Cohen-Addad, C. S. Karthik, and Guillaume Lagarde. On efficient low distortion ultrametric embedding. In Proceedings of the 37th International Conference on Machine Learning, ICML'20. JMLR.org, 2020. Google Scholar
  16. Aaron Coté, Adam Meyerson, and Laura Poplawski. Randomized k-server on hierarchical binary trees. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, pages 227-234, New York, NY, USA, 2008. Association for Computing Machinery. Google Scholar
  17. Sanjoy Dasgupta. A cost function for similarity-based hierarchical clustering. In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 118-127. ACM, 2016. URL: https://doi.org/10.1145/2897518.2897527.
  18. Brian C Dean, Michel X Goemans, and Jan Vondrák. Approximating the stochastic knapsack problem: The benefit of adaptivity. Mathematics of Operations Research, 33(4):945-964, 2008. Google Scholar
  19. Kedar Dhamdhere, Anupam Gupta, and R Ravi. Approximation algorithms for minimizing average distortion. Theory of Computing Systems, 39(1):93-111, 2006. Google Scholar
  20. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences, 69(3):485-497, 2004. Special Issue on STOC 2003. URL: https://doi.org/10.1016/j.jcss.2004.04.011.
  21. Naveen Garg, Goran Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group steiner tree problem. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '98, pages 253-259, USA, 1998. Society for Industrial and Applied Mathematics. Google Scholar
  22. Sudipto Guha and Kamesh Munagala. Approximation algorithms for budgeted learning problems. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 104-113, 2007. Google Scholar
  23. Anupam Gupta, Ilan Newman, Yuri Rabinovich, and Alistair Sinclair. Cuts, trees and 𝓁₁-embeddings of graphs. Combinatorica, 24(2):233-269, April 2004. URL: https://doi.org/10.1007/s00493-004-0015-x.
  24. Alexander Hall and Christos Papadimitriou. Approximating the distortion. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 111-122. Springer, 2005. Google Scholar
  25. Makoto Imase and Bernard M. Waxman. Dynamic steiner tree problem. SIAM Journal on Discrete Mathematics, 4(3):369-384, 1991. URL: https://doi.org/10.1137/0404033.
  26. Piotr Indyk, Avner Magen, Anastasios Sidiropoulos, and Anastasios Zouzias. Online embeddings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 246-259, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. Google Scholar
  27. Jon Kleinberg and Éva Tardos. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and markov random fields. J. ACM, 49(5):616-639, September 2002. Google Scholar
  28. Robert Krauthgamer and Tim Roughgarden. Metric clustering via consistent labeling. In Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '08, pages 809-818, USA, 2008. Society for Industrial and Applied Mathematics. Google Scholar
  29. James R. Lee and Anastasios Sidiropoulos. Pathwidth, trees, and random embeddings. arXiv e-prints, page arXiv:0910.1409, October 2009. URL: https://doi.org/10.48550/arXiv.0910.1409.
  30. Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg, Christos Faloutsos, and Zoubin Ghahramani. Kronecker graphs: An approach to modeling networks. J. Mach. Learn. Res., 11:985-1042, March 2010. Google Scholar
  31. Aurko Roy and Sebastian Pokutta. Hierarchical clustering via spreading metrics. In Proceedings of the 30th International Conference on Neural Information Processing Systems, NIPS'16, pages 2324-2332, Red Hook, NY, USA, 2016. Curran Associates Inc. Google Scholar
  32. Duncan J. Watts and Steven H. Strogatz. Collective dynamics of "small-world" networks. Nature, 393(6684):440-442, 1998. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail