Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Avi Levy
    • 1
  • Harishchandra Ramadas
    • 1
  • Thomas Rothvoss
    • 1
  1. 1.University of WashingtonSeattleUSA

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