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Discrepancy Without Partial Colorings

Authors Nicholas J. A. Harvey, Roy Schwartz, Mohit Singh

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  • Combinatorial Discrepancy
  • Brownian Motion
  • Semi-Definite Programming
  • Randomized Algorithm

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Abstract

Spencer's theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer's theorem that directly computes a coloring, without recursively computing partial colorings.

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Nicholas J. A. Harvey, Roy Schwartz, and Mohit Singh. Discrepancy Without Partial Colorings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 258-273, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.258

Author Details

Nicholas J. A. Harvey
Roy Schwartz
Mohit Singh

References

  1. Noga Alon and Joel Spencer. The Probabilistic Method. John Wiley, 2000. Google Scholar
  2. Wojciech Banaszczyk. Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures & Algorithms, 12(4):351-360, 1998. Google Scholar
  3. N. Bansal and J. Spencer. Deterministic discrepancy minimization. Algorithmica, 67(4):451-471, 2013. Google Scholar
  4. Nikhil Bansal. Constructive algorithms for discrepancy minimization. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 3-10. IEEE, 2010. Google Scholar
  5. Nikhil Bansal, Moses Charikar, Ravishankar Krishnaswamy, and Shi Li. Better algorithms and hardness for broadcast scheduling via a discrepancy approach. In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2014. Google Scholar
  6. József Beck and Tibor Fiala. "Integer-making" theorems. Discrete Applied Mathematics, 3(1):1-8, 1981. Google Scholar
  7. József Beck and Vera T. Sós. Discrepancy theory. In R. Graham and M. Grötschel and L. Lovász, editor, Handbook of Combinatorics, pages 1405-1446. Elsevier Science B.V., 1995. Google Scholar
  8. Stephen P. Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. Google Scholar
  9. Bernard Chazelle. The Discrepancy Method: Randomness and Complexity. Cambridge University Press, 2000. Google Scholar
  10. Friedrich Eisenbrand, Dömötör Pálvölgyi, and Thomas Rothvoß. Bin packing via discrepancy of permutations. ACM Transactions on Algorithms (TALG), 9(3):24, 2013. Google Scholar
  11. Apostolos Giannopoulos. On some vector balancing problems. Studia Mathematica, 122(3):225-234, 1997. Google Scholar
  12. Efim Davydovich Gluskin. Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. Mathematics of the USSR-Sbornik, 64(1):85, 1989. Google Scholar
  13. Shachar Lovett and Raghu Meka. Constructive discrepancy minimization by walking on the edges. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 61-67. IEEE, 2012. Google Scholar
  14. Jiří Matoušek. Geometric discrepancy: An illustrated guide, volume 18. Springer, 1999. Google Scholar
  15. Thomas Rothvoß. Approximating bin packing within O(log OPT ⋅ log log OPT) bins. In Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 20-29. IEEE, 2013. Google Scholar
  16. Thomas Rothvoß. Constructive discrepancy minimization for convex sets. arXiv preprint arXiv:1404.0339, 2014. Google Scholar
  17. Joel Spencer. Six standard deviations suffice. Transactions of the American Mathematical Society, 289(2):679-706, 1985. Google Scholar
  18. Aravind Srinivasan. Improving the discrepancy bound for sparse matrices: better approximations for sparse lattice approximation problems. In Proceedings of the 8th annual ACM-SIAM Symposium on Discrete Algorithms, pages 692-701. Society for Industrial and Applied Mathematics, 1997. Google Scholar
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