Metric extension operators, vertex sparsifiers and Lipschitz extendability

Abstract

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators.

We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of \(\Omega \left( {\sqrt {\log k/\log \log k} } \right)\) for flow sparsifiers and a lower bound of \(\Omega \left( {\sqrt {\log k} /\log \log k} \right)\) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist \(\tilde O\left( {\sqrt {\log k} } \right)\) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than \(\tilde \Omega \left( {\sqrt {\log k} } \right)\) would imply a negative answer to this question.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    S. Arora, J. Lee and A. Naor, Fréchet embeddings of negative type metrics. Discrete & Computational Geometry 38 (2007), 726–739.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    S. Arora, S. Rao and U. Vazirani, Expander flows, geometric embeddings, and graph partitionings, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 222–231.

    Google Scholar 

  3. [3]

    K. Ball, Markov chains, Riesz transforms and Lipschitz maps, Geometric and Functional Analysis 2 (1992), 137–172.

    Article  MATH  Google Scholar 

  4. [4]

    J. Bourgain, A counterexample to a complementation problem, Compositio Mathematica 43 (1981), 133–144.

    MathSciNet  MATH  Google Scholar 

  5. [5]

    G. Calinescu, H. Karloff and Y. Rabani, Approximation algorithms for the 0-extension problem, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), SIAM, Philadelphia, PA, 2001, pp. 8–16.

    Google Scholar 

  6. [6]

    M. Charikar, T. Leighton, S. Li and A. Moitra, Vertex sparsifiers and abstract rounding algorithms, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2010), IEEE Computer Society, Los Alamitos, CA, 2010, pp. 265–274.

    Google Scholar 

  7. [7]

    M. Englert, A. Gupta, R. Krauthgamer, H. Räcke, I. Talgam-Cohen and K. Talwar, Vertex sparsifiers: New results from old techniques, in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 6302, Springer, Berlin, 2010, pp. 152–165.

    Google Scholar 

  8. [8]

    J. Fakcharoenphol, C. Harrelson, S. Rao and K Talwar, An improved approximation algorithm for the 0-extension problem, in Proceedings of the Fourteenth Annual ACMSIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, 2003, pp. 257–265.

    Google Scholar 

  9. [9]

    J. Fakcharoenphol and K. Talwar, An improved decomposition theorem for graphs excluding a fixed minor, in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Computer Science, Vol. 2764, Springer, Berlin, 2003, pp. 36–46.

    Google Scholar 

  10. [10]

    T. Figiel, W. Johnson and G. Schechtman, Factorizations of natural embeddings of ln p into L r , I, Studia Mathematica 89 (1988), 79–103.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    B. Grünbaum, Projection constants, Transactions of the American Mathematical Society, 95 (1960), 451–465.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    U. Haagerup, The best constants in the Khintchine inequality, Studia Mathematica 70 (1981), 231–283.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    W. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space, in Conference in Modern Analysis and Probability (New Haven, Conn., 1982), Contemporary Mathematics, Vol. 26, American Mathematical Society, Providence, RI, 1984, pp. 189–206.

    Google Scholar 

  14. [14]

    A. Karzanov, Minimum 0-extension of graph metrics, European Journal of Combinatorics 19 (1998), 71–101.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    B. Kashin, The widths of certain finite-dimensional sets and classes of smooth functions, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 41 (1977), 334–351.

    MathSciNet  Google Scholar 

  16. [16]

    M. D. Kirszbraun, Über die zusammenziehenden und Lipschitzchen Transformationen, Fundamenata Mathematicae 22 (1934), 77–108.

    MATH  Google Scholar 

  17. [17]

    J. Lee and A. Naor, Extending Lipschitz functions via random metric partitions, Inventiones Mathematicae 160 (2005), 59–95.

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    J. Lee and A. Sidiropoulos, Genus and the geometry of the cut graph, in Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms (Austin, TX, 2010) SIAM, Philadelphia, PA, 2010, pp. 193–201.

    Google Scholar 

  19. [19]

    T. Leighton and A. Moitra, Extensions and limits to vertex sparsification, in Proceedings of the 2010 ACM International Symposium on Theory of Computing, ACM, New York, 2010, pp. 47–56.

    Google Scholar 

  20. [20]

    M. B. Marcus and G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Mathematica 152 (1984), 245–301.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    E. J. McShane, Extension of range of functions, Bulletin of the American Mathematical Society 40 (1934), 837–842.

    MathSciNet  Article  MATH  Google Scholar 

  22. [22]

    M. Mendel and A. Naor, Some applications of Ball’s extension theorem, Proceedings of the American Mathematical Society 134 (2006), 2577–2584.

    MathSciNet  Article  MATH  Google Scholar 

  23. [23]

    A. Moitra, Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size, in Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (Atlanta, GA, 2009) IEEE Computer Society, Los Alamitos, CA, 2009, pp. 3–12.

    Google Scholar 

  24. [24]

    A. Naor, Y. Peres, O. Schrammand S. Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Mathematical Journal 134 (2006), 165–197.

    MathSciNet  Article  MATH  Google Scholar 

  25. [25]

    J. von Neumann, Zur Theorie der Gesellshaftsphiele, Mathematische Annalen 100 (1928), 295–320.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    H. Räcke, Optimal hierarchical decompositions for congestion minimization in networks, in Proceedings of the 40th Annual ACM Symposium on Theory of Computing (Victoria, BC, 2008), ACM, New York, 2008, pp. 255–264.

    Google Scholar 

  27. [27]

    B. Randrianantoanina, Extensions of Lipschitz maps, International Conference on Banach Spaces and Operator Spaces, 2007.

    Google Scholar 

  28. [28]

    A. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, Vol. 24, Springer, Berlin, 2003.

    Google Scholar 

  29. [29]

    M. Sion, On general minimax theorems, Pacific Journal of Mathematics 8 (1958), 171–176.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Konstantin Makarychev or Yury Makarychev.

Additional information

The conference version of this paper appeared at FOCS 2010.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Makarychev, K., Makarychev, Y. Metric extension operators, vertex sparsifiers and Lipschitz extendability. Isr. J. Math. 212, 913–959 (2016). https://doi.org/10.1007/s11856-016-1315-8

Download citation