Optimal Lower Bounds for Universal and Differentially Private Steiner Trees and TSPs

  • Anand Bhalgat
  • Deeparnab Chakrabarty
  • Sanjeev Khanna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6845)

Abstract

Given a metric space on n points, an α-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an α factor of the optimal Steiner tree cost for X. An α-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an α factor of the optimal as before, and satisfies the additional property that for any set X′ that differs in a single vertex from X, the tree distributions for X and X′ are “close” to each other. Universal and differentially private algorithms for TSP are defined similarly. An α-approximate universal algorithm for the Steiner tree problem or TSP is also an α-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(logn) approximate differentially private algorithms as well.

We prove an Ω(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root. We then show that whenever the universal problem has a lower bound that satisfies an additional property, it implies a similar lower bound for the differentially private version. Using this converse relation between universal and private algorithms, we establish an Ω(logn) lower bound for the differentially private Steiner tree and the differentially private TSP. This answers a question of Talwar [19]. Our results highlight a natural connection between universal and private approximation algorithms that is likely to have other applications.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Archer, A.: Two O(log* k)-approximation algorithms for the asymmetric k-center problem. In: Proceedings, MPS Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 1–14 (2010)Google Scholar
  2. 2.
    Bartal, Y.: On approximating arbitrary metrices by tree metrics. In: ACM Symp. on Theory of Computing (STOC), pp. 161–168 (1998)Google Scholar
  3. 3.
    Bhalgat, A., Chakrabarty, D., Khanna, S.: Optimal lower bounds for universal and differentially private steiner trees and tsps. Technical report, http://arxiv.org/abs/1011.3770
  4. 4.
    Dwork, C.: Differential privacy. In: Proceedings, International Colloquium on Automata, Languages and Processing, pp. 1–12 (2006)Google Scholar
  5. 5.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: ACM Symp. on Theory of Computing (STOC), pp. 448–455 (2003)Google Scholar
  6. 6.
    Gorodezky, I., Kleinberg, R.D., Shmoys, D.B., Spencer, G.: Improved lower bounds for the universal and a priori tsp. In: Proceedings, International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp. 178–191 (2010)Google Scholar
  7. 7.
    Gupta, A., Hajiaghayi, M., Räcke, H.: Oblivious network design. In: Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 970–979 (2006)Google Scholar
  8. 8.
    Gupta, A., Ligett, K., McSherry, F., Roth, A., Talwar, K.: Differentially private approximation algorithms. In: Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1106–1125 (2010)Google Scholar
  9. 9.
    Hajiaghayi, M., Kleinberg, R., Leighton, F.T.: Improved lower and upper bounds for universal tsp in planar metrics. In: Proceedings, ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 649–658 (2006)Google Scholar
  10. 10.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. of the Amer. Soc. 43(4), 439–561 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Imase, M., Waxman, B.M.: Dynamic steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jia, L., Lin, G., Noubir, G., Rajaraman, R., Sundaram, R.: Universal approximations for tsp, steiner tree, and set cover. In: ACM Symp. on Theory of Computing (STOC), pp. 386–395 (2005)Google Scholar
  13. 13.
    Leighton, F.T., Rao, S.: An approximate max-flow min-cut theorem for uniform multicommodity flow problems with application to approximation algorithms. In: Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pp. 422–431 (1988)Google Scholar
  14. 14.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–246 (1995)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 4, 261–277 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Moitra, A.: Approximation algorithms for multicommodity-type problems with guarantees independent of the graph size. In: Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pp. 3–12 (2009)Google Scholar
  17. 17.
    Moitra, A., Leighton, F.T.: Extensions and limits to vertex sparsification. In: ACM Symp. on Theory of Computing (STOC), pp. 47–56 (2010)Google Scholar
  18. 18.
    Panigrahy, R., Vishwanathan, S.: An O(log* n) approximation algorithm for the asymmetric p-center problem. J. Algorithms 27(2), 259–268 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Talwar, K.: Problem 1. In: Open Problem in Bellairs Workshop on Approximation Algorithms, Barbados (2010), http://www.math.mcgill.ca/~vetta/Workshop/openproblems2.pdf
  20. 20.
    Yao, A.C.-C.: Probabilistic computations: Towards a unified measure of complexity. In: Proceedings, IEEE Symposium on Foundations of Computer Science (FOCS), pp. 222–227 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Anand Bhalgat
    • 1
  • Deeparnab Chakrabarty
    • 1
  • Sanjeev Khanna
    • 1
  1. 1.Department of Computer and Information ScienceUniversity of PennsylvaniaUSA

Personalised recommendations