STACS 2007: STACS 2007 pp 441-452 | Cite as

Randomly Rounding Rationals with Cardinality Constraints and Derandomizations

  • Benjamin Doerr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

We show how to generate randomized roundings of rational vectors that satisfy hard cardinality constraints and allow large deviations bounds. This improves and extends earlier results by Srinivasan (FOCS 2001), Gandhi et al. (FOCS 2002) and the author (STACS 2006). Roughly speaking, we show that also for rounding arbitrary rational vectors randomly or deterministically, it suffices to understand the problem for \(\{0,\tfrac 12\}\) vectors (which typically is much easier). So far, this was only known for vectors with entries in 2 − ℓ ℤ, ℓ ∈ ℕ.

To prove the general case, we exhibit a number of results of independent interest, in particular, a quite useful lemma on negatively correlated random variables, an extension of de Werra’s (RAIRO 1971) coloring result for unimodular hypergraphs and a sufficient condition for a unimodular hypergraph to have a perfectly balanced non-trivial partial coloring.

We also show a new solution for the general derandomization problem for rational matrices.

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References

  1. [AS04]
    Ageev, A.A., Sviridenko, M.I.: Pipage rounding: a new method of constructing algorithms with proven performance guarantee. J. Comb. Optim. 8, 307–328 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. [DGS05]
    Doerr, B., Gnewuch, M., Srivastav, A.: Bounds and constructions for the star-discrepancy via δ-covers. Journal of Complexity 21, 691–709 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. [DK06]
    Doerr, B., Klein, C.: Unbiased rounding of rational matrices. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 200–211. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. [Doe03]
    Doerr, B.: Non-independent randomized rounding. In: SODA 2003, pp. 506–507 (2003)Google Scholar
  5. [Doe04a]
    Doerr, B.: Global roundings of sequences. Information Processing Letters 92, 113–116 (2004)CrossRefMathSciNetGoogle Scholar
  6. [Doe04b]
    Doerr, B.: Nonindependent randomized rounding and an application to digital halftoning. SIAM Journal on Computing 34, 299–317 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. [Doe06]
    Doerr, B.: Generating randomized roundings with cardinality constraints and derandomizations. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 571–583. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. [dW71]
    de Werra, D.: Equitable colorations of graphs. Rev. Française Informat. Recherche Opérationnelle 5(Ser. R-3), 3–8 (1971)Google Scholar
  9. [GHK+06]
    Gandhi, R., et al.: An improved approximation algorithm for vertex cover with hard capacities. J. Comput. Syst. Sci. 72, 16–33 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. [GKPS02]
    Gandhi, R., et al.: Dependent rounding in bipartite graphs. In: FOCS 2002, pp. 323–332 (2002)Google Scholar
  11. [PS97]
    Panconesi, A., Srinivasan, A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput. 26, 350–368 (1997)MATHCrossRefMathSciNetGoogle Scholar
  12. [Rag88]
    Raghavan, P.: Probabilistic construction of deterministic algorithms: Approximating packing integer programs. J. Comput. Syst. Sci. 37, 130–143 (1988)MATHCrossRefMathSciNetGoogle Scholar
  13. [RT87]
    Raghavan, P., Thompson, C.D.: Randomized rounding: A technique for provably good algorithms and algorithmic proofs. Combinatorica 7, 365–374 (1987)MATHCrossRefMathSciNetGoogle Scholar
  14. [RT91]
    Raghavan, P., Thompson, C.D.: Multiterminal global routing: a deterministic approximation scheme. Algorithmica 6, 73–82 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. [Sri01a]
    Srinivasan, A.: Distributions on level-sets with applications to approximations algorithms. In: FOCS 2001, pp. 588–597 (2001)Google Scholar
  16. [SS96]
    Srivastav, A., Stangier, P.: Algorithmic Chernoff-Hoeffding inequalities in integer programming. Random Structures & Algorithms 8, 27–58 (1996)MATHCrossRefMathSciNetGoogle Scholar
  17. [STT01]
    Sadakane, K., Takki-Chebihi, N., Tokuyama, T.: Combinatorics and algorithms on low-discrepancy roundings of a real sequence. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 166–177. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Benjamin Doerr
    • 1
  1. 1.Max–Planck–Institut für Informatik, SaarbrückenGermany

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