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Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 10328)

Abstract

A well-known theorem of Spencer shows that any set system with n sets over n elements admits a coloring of discrepancy \(O(\sqrt{n})\). While the original proof was non-constructive, recent progress brought polynomial time algorithms by Bansal, Lovett and Meka, and Rothvoss. All those algorithms are randomized, even though Bansal’s algorithm admitted a complicated derandomization.

We propose an elegant deterministic polynomial time algorithm that is inspired by Lovett-Meka as well as the Multiplicative Weight Update method. The algorithm iteratively updates a fractional coloring while controlling the exponential weights that are assigned to the set constraints.

A conjecture by Meka suggests that Spencer’s bound can be generalized to symmetric matrices. We prove that \(n \times n\) matrices that are block diagonal with block size q admit a coloring of discrepancy \(O(\sqrt{n} \cdot \sqrt{\log (q)})\). Bansal, Dadush and Garg recently gave a randomized algorithm to find a vector x with entries in \(\lbrace {-1,1\rbrace }\) with \(\Vert Ax\Vert _{\infty } \le O(\sqrt{\log n})\) in polynomial time, where A is any matrix whose columns have length at most 1. We show that our method can be used to deterministically obtain such a vector.

Keywords

  • Symmetric Matrix
  • Polynomial Time Algorithm
  • Symmetric Matrice
  • Semidefinite Program
  • Exponential Weight

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

T. Rothvoss—Supported by NSF grant 1420180 with title “Limitations of convex relaxations in combinatorial optimization”, an Alfred P. Sloan Research Fellowship and a David & Lucile Packard Foundation Fellowship.

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Notes

  1. 1.

    We should mention for the sake of completeness that our update choice is not a convex combination of the experts weighted by their exponential weights.

  2. 2.

    See the blog post https://windowsontheory.org/2014/02/07/discrepancy-and-be ating-the-union-bound/.

  3. 3.

    See https://arxiv.org/abs/1611.08752.

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Correspondence to Harishchandra Ramadas .

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Levy, A., Ramadas, H., Rothvoss, T. (2017). Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_31

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  • DOI: https://doi.org/10.1007/978-3-319-59250-3_31

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