On Quadratic Programming with a Ratio Objective

  • Aditya Bhaskara
  • Moses Charikar
  • Rajsekar Manokaran
  • Aravindan Vijayaraghavan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

Quadratic Programming (QP) is the well-studied problem of maximizing over { − 1,1} values the quadratic form ∑ i ≠ jaijxixj. QP captures many known combinatorial optimization problems, and assuming the Unique Games conjecture, Semidefinite Programming (SDP) techniques give optimal approximation algorithms. We extend this body of work by initiating the study of Quadratic Programming problems where the variables take values in the domain { − 1,0,1}. The specific problem we study is
$$\begin{aligned} \textsf{QP-Ratio} &: \mbox{\ \ } \max_{\{-1,0,1\}^n} \frac{\sum_{i \not = j} a_{ij} x_i x_j}{\sum x_i^2} \end{aligned}$$
This is a natural relative of several well studied problems (in fact Trevisan introduced a normalized variant as a stepping stone towards a spectral algorithm for Max Cut Gain). Quadratic ratio problems are good testbeds for both algorithms and complexity because the techniques used for quadratic problems for the { − 1,1} and {0,1} domains do not seem to carry over to the { − 1,0,1} domain. We give approximation algorithms and evidence for the hardness of approximating these problems.

We consider an SDP relaxation obtained by adding constraints to the natural eigenvalue (or SDP) relaxation for this problem. Using this, we obtain an \(\tilde{O}(n^{1/3})\) approximation algorithm for QP-ratio. We also give a \(\tilde{O}(n^{1/4})\) approximation for bipartite graphs, and better algorithms for special cases.

As with other problems with ratio objectives (e.g. uniform sparsest cut), it seems difficult to obtain inapproximability results based on P ≠ NP. We give two results that indicate that QP-Ratio is hard to approximate to within any constant factor: one is based on the assumption that random instances of Max k-AND are hard to approximate, and the other makes a connection to a ratio version of Unique Games. We also give a natural distribution on instances of QP-Ratio for which an nε approximation (for ε roughly 1/10) seems out of reach of current techniques.

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References

  1. [ABH+05]
    Arora, S., Berger, E., Hazan, E., Kindler, G., Safra, M.: On non-approximability for quadratic programs. In: FOCS 2005: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 206–215. IEEE Computer Society, Washington, DC (2005)Google Scholar
  2. [AN06]
    Alon, N., Naor, A.: Approximating the cut-norm via grothendieck’s inequality. SIAM J. Comput. 35, 787–803 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. [Aus07]
    Austrin, P.: Towards sharp inapproximability for any 2-csp. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 307–317. IEEE Computer Society, Washington, DC (2007)Google Scholar
  4. [BCC+10]
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an o(n 1/4) approximation for densest k-subgraph. In: STOC 2010: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 201–210. ACM, New York (2010)CrossRefGoogle Scholar
  5. [BCMV11]
    Bhaskara, A., Charikar, M., Manokaran, R., Vijayaraghavan, A.: On quadratic programming with a ratio objective. CoRR, abs/1101.1710 (2011)Google Scholar
  6. [Cha00]
    Charikar, M.: Greedy Approximation Algorithms for Finding Dense Components in a Graph. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 84–95. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. [CMM06]
    Charikar, M., Makarychev, K., Makarychev, Y.: Near-optimal algorithms for unique games. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC 2006, pp. 205–214. ACM, New York (2006)CrossRefGoogle Scholar
  8. [CW04]
    Charikar, M., Wirth, A.: Maximizing quadratic programs: Extending grothendieck’s inequality. In: FOCS 2004: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 54–60. IEEE Computer Society, Washington, DC (2004)CrossRefGoogle Scholar
  9. [DKS11]
    Deshpande, A., Kannan, R., Srivastava, N.: Zero-one rounding of singular vectors. Manuscript (2011)Google Scholar
  10. [Fei02]
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of the 34th annual ACM Symposium on Theory of Computing (STOC 2002), pp. 534–543. ACM Press (2002)Google Scholar
  11. [GW95]
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42(6), 1115–1145 (1995)MathSciNetMATHCrossRefGoogle Scholar
  12. [KKMO07]
    Khot, S., Kindler, G., Mossel, E., O’Donnell, R.: Optimal inapproximability results for max-cut and other 2-variable csps? SIAM J. Comput. 37(1), 319–357 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. [KV05]
    Khot, S., Vishnoi, N.K.: The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into ℓ1. In: FOCS 2005, pp. 53–62 (2005)Google Scholar
  14. [NRT99]
    Nemirovski, A., Roos, C., Terlaky, T.: On maximization of quadratic form over intersection of ellipsoids with common center. Mathematical Programming 86, 463–473 (1999), doi:10.1007/s101070050100MathSciNetMATHCrossRefGoogle Scholar
  15. [Rag08]
    Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: STOC 2008, pp. 245–254 (2008)Google Scholar
  16. [RS10]
    Raghavendra, P., Steurer, D.: Graph expansion and the unique games conjecture. In: STOC 2010: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 755–764. ACM, New York (2010)CrossRefGoogle Scholar
  17. [Tre09]
    Trevisan, L.: Max cut and the smallest eigenvalue. In: STOC 2009: Proceedings of the 41st annual ACM Symposium on Theory of Computing, pp. 263–272. ACM, New York (2009)CrossRefGoogle Scholar
  18. [Tul09]
    Tulsiani, M.: Csp gaps and reductions in the lasserre hierarchy. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 303–312. ACM, New York (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aditya Bhaskara
    • 1
  • Moses Charikar
    • 1
  • Rajsekar Manokaran
    • 1
  • Aravindan Vijayaraghavan
    • 1
  1. 1.Department of Computer SciencePrinceton University, Center for Computational IntractabilityUSA

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