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A semidefinite programming hierarchy for packing problems in discrete geometry

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Abstract

Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre’s semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.

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Notes

  1. Assume \(\theta _1 < \theta _2\) and let \(\epsilon \) be some number strictly between \((1 - \theta _1/\theta _2)/2\) and \(1\). Let \(D = (0, 1)\), and let \(d((x,i), (y,j))\) be given by \(\epsilon \delta _{i \ne j} + (1 - \epsilon \delta _{i \ne j}) \arccos (x \cdot y)\, (\theta _1+\theta _2)^{-1}\) when \(x \cdot y < \cos (\theta _i + \theta _j)\) and \(1\) otherwise.

  2. Consider the graph with vertex set \([0, 1] \times \mathbb {Z}\) where \((x, i)\) and \((y, j)\) are adjacent if \(i = j\) or when \(x\) and \(y\) are both strictly smaller than \(|i-j|^{-1}\) (for \(i \ne j\)). Here each finite clique is contained in an open clique, but the countable clique \(\{0\} \times \mathbb {Z}\) is not.

  3. In this paper cones are always assumed to be convex.

  4. We show this in Remark 3.

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Acknowledgments

We would like to thank Evan DeCorte and Cristóbal Guzmán for very helpful discussions. We also thank the referee whose suggestions helped to improve the paper.

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Authors and Affiliations

  1. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA , Delft, The Netherlands

    David de Laat

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931, Cologne, Germany

    Frank Vallentin

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  1. David de Laat
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  2. Frank Vallentin
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Correspondence to David de Laat.

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The authors were supported by Vidi Grant 639.032.917 from the Netherlands Organization for Scientific Research (NWO).

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de Laat, D., Vallentin, F. A semidefinite programming hierarchy for packing problems in discrete geometry. Math. Program. 151, 529–553 (2015). https://doi.org/10.1007/s10107-014-0843-4

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