Advertisement

Algorithmica

, Volume 80, Issue 12, pp 3803–3824 | Cite as

Quasimetric Embeddings and Their Applications

  • Facundo Mémoli
  • Anastasios Sidiropoulos
  • Vijay SridharEmail author
Article
  • 88 Downloads

Abstract

We study generalizations of classical metric embedding results to the case of quasimetric spaces; that is, spaces that do not necessarily satisfy symmetry. Quasimetric spaces arise naturally from the shortest-path distances on directed graphs. Perhaps surprisingly, very little is known about low-distortion embeddings for quasimetric spaces.Random embeddings into ultrametric spaces are arguably one of the most successful geometric tools in the context of algorithm design. We extend this to the quasimetric case as follows. We show that any n-point quasimetric space supported on a graph of treewidth t admits a random embedding into quasiultrametric spaces with distortion \(O(t \log ^2 n)\), where quasiultrametrics are a natural generalization of ultrametrics. This result allows us to obtain \(t\log ^{O(1)} n\)-approximation algorithms for the Directed Non-Bipartite Sparsest Cut and the Directed Multicut problems on n-vertex graphs of treewidth t, with running time polynomial in both n and t. The above results are obtained by considering a generalization of random partitions to the quasimetric case, which we refer to as random quasipartitions. Using this definition and a construction of Chuzhoy and Khanna (JACM 56(2):6, 2009) we derive a polynomial lower bound on the distortion of random embeddings of general quasimetric spaces into quasiultrametric spaces. Finally, we establish a lower bound for embedding the shortest-path quasimetric of a graph G into graphs that exclude G as a minor. This lower bound is used to show that several embedding results from the metric case do not have natural analogues in the quasimetric setting.

Keywords

Metric embeddings Quasimetrics Outliers Random embeddings Treewidth Directed Sparsest-Cut Directed Multicut 

References

  1. 1.
    Abraham, I., Bartal, Y., Neiman, O.: Nearly tight low stretch spanning trees. arXiv preprint arXiv:0808.2017 (2008)
  2. 2.
    Agarwal, A., Alon, N., Charikar, M.S.: Improved approximation for directed cut problems. In: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 671–680. ACM (2007)Google Scholar
  3. 3.
    Arora, S., Lee, J., Naor, A.: Euclidean distortion and the sparsest cut. J. Am. Math. Soc. 21(1), 1–21 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. J. ACM 56(2), 5 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bartal, Y.: Probabilistic approximation of metric spaces and its algorithmic applications. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, 1996, pp. 184–193. IEEE (1996)Google Scholar
  6. 6.
    Bartal, Y.: On approximating arbitrary metrices by tree metrics. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 161–168. ACM (1998)Google Scholar
  7. 7.
    Borradaile, G., Lee, J.R., Sidiropoulos, A.: Randomly removing g handles at once. In: Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, pp. 371–376. ACM (2009)Google Scholar
  8. 8.
    Calinescu, G., Karloff, H., Rabani, Y.: Approximation algorithms for the 0-extension problem. SIAM J. Comput. 34(2), 358–372 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Charikar, M., Makarychev, K., Makarychev, Y.: Directed metrics and directed graph partitioning problems. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, pp. 51–60. Society for Industrial and Applied Mathematics (2006)Google Scholar
  10. 10.
    Chawla, S., Krauthgamer, R., Kumar, R., Rabani, Y., Sivakumar, D.: On the hardness of approximating multicut and sparsest-cut. Comput. Complex. 15(2), 94–114 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cheriyan, J., Karloff, H., Rabani, Y.: Approximating directed multicuts. Combinatorica 25(3), 251–269 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chuzhoy, J., Khanna, S.: Hardness of cut problems in directed graphs. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 527–536. ACM (2006)Google Scholar
  13. 13.
    Chuzhoy, J., Khanna, S.: Polynomial flow-cut gaps and hardness of directed cut problems. J. ACM 56(2), 6 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 448–455. ACM (2003)Google Scholar
  15. 15.
    Feige, U., Hajiaghayi, M., Lee, J.R.: Improved approximation algorithms for minimum weight vertex separators. SIAM J. Comput. 38(2), 629–657 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hajiaghayi, M.T., Räcke, H.: An-approximation algorithm for directed sparsest cut. Inf. Process. Lett. 97(4), 156–160 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Indyk, P., Matoušek, J.: Low-distortion embeddings of finite metric spaces. In: Handbook of Discrete and Computational Geometry, p. 177 (2004)Google Scholar
  18. 18.
    Indyk, P., Sidiropoulos, A.: Probabilistic embeddings of bounded genus graphs into planar graphs. In: Proceedings of the Twenty-Third Annual Symposium on Computational Geometry, pp. 204–209. ACM (2007)Google Scholar
  19. 19.
    Kelner, J.A., Lee, J.R., Price, G.N., Teng, S.H.: Metric uniformization and spectral bounds for graphs. Geom. Funct. Anal. 21(5), 1117–1143 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 767–775. ACM (2002)Google Scholar
  21. 21.
    Khot, S.A., Vishnoi, N.K.: The unique games conjecture, integrality gap for cut problems and embeddability of negative-type metrics into 1. J. ACM 62(1), 8 (2015)CrossRefGoogle Scholar
  22. 22.
    Klein, P., Plotkin, S.A., Rao, S.: Excluded minors, network decomposition, and multicommodity flow. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp. 682–690. ACM (1993)Google Scholar
  23. 23.
    Kortsarts, Y., Kortsarz, G., Nutov, Z.: Greedy approximation algorithms for directed multicuts. Networks 45(4), 214–217 (2005)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Krauthgamer, R., Lee, J.R., Mendel, M., Naor, A.: Measured descent: a new embedding method for finite metrics. Geom. Funct. Anal. GAFA 15(4), 839–858 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lee, J.R., Naor, A.: Extending lipschitz functions via random metric partitions. Invent. Math. 160(1), 59–95 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lee, J.R., Sidiropoulos, A.: On the geometry of graphs with a forbidden minor. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 245–254. ACM (2009)Google Scholar
  27. 27.
    Lee, J.R., Sidiropoulos, A.: Genus and the geometry of the cut graph. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 193–201. Society for Industrial and Applied Mathematics (2010)Google Scholar
  28. 28.
    Leighton, T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In: 29th Annual Symposium on Foundations of Computer Science, 1988, pp. 422–431. IEEE (1988)Google Scholar
  29. 29.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sidiropoulos, A.: Optimal stochastic planarization. In: 2010 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 163–170. IEEE (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

Personalised recommendations