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Mathematical Programming

, Volume 151, Issue 2, pp 529–553 | Cite as

A semidefinite programming hierarchy for packing problems in discrete geometry

  • David de Laat
  • Frank Vallentin
Full Length Paper Series B

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.

Keywords

Lasserre hierarchy Weighted independence number (stability number) Infinite graphs Geometric packing problems Moment measures 

Mathematics Subject Classification

90C22 52C17 

Notes

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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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands
  2. 2.Mathematisches InstitutUniversität zu KölnCologneGermany

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