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Algorithmic Aspects of Combinatorial Discrepancy

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Book cover A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

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Abstract

This chapter describes some recent results in combinatorial discrepancy theory motivated by designing efficient polynomial time algorithms for finding low discrepancy colorings. Until recently, the best known results for several combinatorial discrepancy problems were based on counting arguments, most notably the entropy method, and were widely believed to be non-algorithmic. We describe some algorithms based on semidefinite programming that can achieve many of these bounds. Interestingly, the new connections between semidefinite optimization and discrepancy have lead to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound and improved bounds on the discrepancy of union of set systems. We will also see a surprising new algorithmic proof of Spencer’s celebrated six standard deviations result due to Lovett and Meka, that does not rely on any semidefinite programming or counting argument.

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Notes

  1. 1.

    Interestingly, an improved \(O(\sqrt{t\log n})\) bound is also known [2] using a different method based on convex geometry.

  2. 2.

    Observe that proving a c-approximation for a problem, implies that the (approximate) problem is both in NP and co-NP. Note that both Theorems 6 and 8 can be used to give a co-NP witness.

  3. 3.

    Note that one cannot set \(\lambda = o(\sqrt{n})\) in our setting, there are set systems on m = O(n) sets (e.g. the Hadamard set system, that we will see in Sect. 6.6) with vector discrepancy \(\varOmega (\sqrt{n})\).

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Authors and Affiliations

  1. Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands

    Nikhil Bansal

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  1. Nikhil Bansal
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Correspondence to Nikhil Bansal .

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Editors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

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Bansal, N. (2014). Algorithmic Aspects of Combinatorial Discrepancy. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_6

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