skip to main content
10.1145/2488608.2488610acmconferencesArticle/Chapter ViewAbstractPublication PagesstocConference Proceedingsconference-collections
research-article

A PRG for lipschitz functions of polynomials with applications to sparsest cut

Published:01 June 2013Publication History

ABSTRACT

We give improved pseudorandom generators (PRGs) for Lipschitz functions of low-degree polynomials over the hypercube. These are functions of the form ψ(P(x)), where P:{1,-1}n -> R is a low-degree polynomial and ψ:R -> R is a function with small Lipschitz constant. PRGs for smooth functions of low-degree polynomials have received a lot of attention recently and play an important role in constructing PRGs for the natural class of polynomial threshold functions [12,13,24,16,15]. In spite of the recent progress, no nontrivial PRGs were known for fooling Lipschitz functions of degree O(log n) polynomials even for constant error rate. In this work, we give the first such generator obtaining a seed-length of (log n)~O(l22) for fooling degree l polynomials with error ε. Previous generators had an exponential dependence on the degree l. We use our PRG to get better integrality gap instances for sparsest cut, a fundamental problem in graph theory with many applications in graph optimization. We give an instance of uniform sparsest cut for which a powerful semi-definite relaxation (SDP) first introduced by Goemans and Linial and studied in the seminal work of Arora, Rao and Vazirani [3] has an integrality gap of exp(Ω((log log n)1/2)). Understanding the performance of the Goemans-Linial SDP for uniform sparsest cut is an important open problem in approximation algorithms and metric embeddings. Our work gives a near-exponential improvement over previous lower bounds which achieved a gap of Ω(log log n) [11,21]. Our gap instance builds on the recent short code gadgets of Barak et al. [5].

References

  1. N. Alon and J.H. Spencer. The Probabilistic Method. John Wiley & Sons, 2000.Google ScholarGoogle ScholarCross RefCross Ref
  2. Sanjeev Arora, James Lee, and Assaf Naor. Euclidean distortion and the sparsest cut. In STOC, pages 553--562, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Sanjeev Arora, Satish Rao, and Umesh V. Vazirani. Expander flows, geometric embeddings and graph partitioning. J. ACM, 56(2), 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. James Aspnes, Richard Beigel, Merrick L. Furst, and Steven Rudich. The expressive power of voting polynomials. Combinatorica, 14(2):135--148, 1994.Google ScholarGoogle ScholarCross RefCross Ref
  5. Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. Making the long code shorter, with applications to the unique games conjecture. In FOCS, 2012. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. Richard Beigel. The polynomial method in circuit complexity. In Proc. of $8$th Annual Structure in Complexity Theory Conference, pages 82--95, 1993.Google ScholarGoogle ScholarCross RefCross Ref
  7. Larry Carter and Mark N. Wegman. Universal classes of hash functions. In STOC, pages 106--112, 1977. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94--114, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Jeff Cheeger, Bruce Kleiner, and Assaf Naor. A (log n)Ω(1) integrality gap for the sparsest cut SDP. In FOCS, pages 555--564, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Eden Chlamtac and Madhur Tulsiani. Handbook on Semidefinite, Conic and Polynomial Optimization, chapter Convex Relaxations and Integrality Gaps. Springer, 2011.Google ScholarGoogle Scholar
  11. Nikhil R. Devanur, Subhash Khot, Rishi Saket, and Nisheeth K. Vishnoi. Integrality gaps for sparsest cut and minimum linear arrangement problems. In STOC, pages 537--546, 2006. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. In FOCS, pages 171--180, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Ilias Diakonikolas, Daniel M. Kane, and Jelani Nelson. Bounded independence fools degree-2 threshold functions. In FOCS, pages 11--20, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Michel X. Goemans. Semidefinite programming in combinatorial optimization. Math. Program., 79:143--161, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Daniel M. Kane. k-independent Gaussians fool polynomial threshold functions. In IEEE Conference on Computational Complexity, pages 252--261, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Daniel M. Kane. A small PRG for polynomial threshold functions of Gaussians. In FOCS, pages 257--266, 2011. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Daniel M. Kane. A structure theorem for poorly anticoncentrated gaussian chaoses and applications to the study of polynomial threshold functions. In FOCS, pages 91--100, 2012. Google ScholarGoogle ScholarDigital LibraryDigital Library
  18. Subhash Khot and Rishi Saket. SDP integrality gaps with local l_1-embeddability. In FOCS, pages 565--574, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Subhash Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l_1. In FOCS, pages 53--62, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Adam R. Klivans and Rocco A. Servedio. Learning DNF in time 2O(n1/3). J. Comp. Sys. Sc., 68(2):303--318, 2004. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Robert Krauthgamer and Yuval Rabani. Improved lower bounds for embeddings into $\ell_1$. SIAM J. Comput., 38(6):2487--2498, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Nathan Linial. Finite metric spaces: combinatorics, geometry and algorithms. In Proceedings of the eighteenth annual symposium on Computational geometry, SCG '02, pages 63--63, 2002. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215--245, 1995.Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Raghu Meka and David Zuckerman. Pseudorandom generators for polynomial threshold functions. In STOC, pages 427--436, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences invariance and optimality. In FOCS, pages 21--30, 2005. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Yuval Rabani and Amir Shpilka. Explicit construction of a small epsilon-net for linear threshold functions. SIAM J. Comput., 39(8):3501--3520, 2010. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Yuri Rabinovich. On average distortion of embedding metrics into the line and into l1. In STOC, pages 456--462, 2003. Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Prasad Raghavendra and David Steurer. Integrality gaps for strong SDP relaxations of unique games. In FOCS, pages 575--585, 2009. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Farhad Shahrokhi and David W. Matula. The maximum concurrent flow problem. J. ACM, 37(2):318--334, 1990. Google ScholarGoogle ScholarDigital LibraryDigital Library
  30. David B. Shmoys. Approximation algorithms for np-hard problems. chapter Cut problems and their application to divide-and-conquer, pages 192--235. PWS Publishing Co., Boston, MA, USA, 1997. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. A PRG for lipschitz functions of polynomials with applications to sparsest cut

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing
        June 2013
        998 pages
        ISBN:9781450320290
        DOI:10.1145/2488608

        Copyright © 2013 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 June 2013

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article

        Acceptance Rates

        STOC '13 Paper Acceptance Rate100of360submissions,28%Overall Acceptance Rate1,469of4,586submissions,32%

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!