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Deterministic Discrepancy Minimization

Abstract

We derandomize a recent algorithmic approach due to Bansal (Foundations of Computer Science, FOCS, pp. 3–10, 2010) to efficiently compute low discrepancy colorings for several problems, for which only existential results were previously known. In particular, we give an efficient deterministic algorithm for Spencer’s six standard deviations result (Spencer in Trans. Am. Math. Soc. 289:679–706, 1985), and to find a low discrepancy coloring for a set system with low hereditary discrepancy.

The main new idea is to add certain extra constraints to the natural semidefinite programming formulation for discrepancy, which allow us to argue about the existence of a good deterministic move at each step of the algorithm. The non-constructive entropy method is used to argue the feasibility of this enhanced SDP.

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Notes

  1. 1.

    For general m, we can also obtain the bound \(O(\sqrt{n} \log(m/n))\) as in [2], but to we restrict our attention here to m=O(n) which is the most interesting case.

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Author information

Correspondence to Nikhil Bansal.

Additional information

A preliminary version of this paper appeared in European Symposium on Algorithms, ESA 2011.

This research has been supported by the Netherlands Organisation for Scientific Research (NWO) grant 639.022.211.

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Bansal, N., Spencer, J. Deterministic Discrepancy Minimization. Algorithmica 67, 451–471 (2013). https://doi.org/10.1007/s00453-012-9728-1

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Keywords

  • Discrepancy
  • Derandomization
  • Algorithms
  • Semi-definite programming