Mathematical Programming

, Volume 154, Issue 1–2, pp 407–425 | Cite as

Lower bounds on the sizes of integer programs without additional variables

  • Volker Kaibel
  • Stefan Weltge
Full Length Paper Series B


Let \( X \) be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is \( X \). This quantity, which we call the relaxation complexity of \( X \), corresponds to the smallest number of linear inequalities of any integer program having \( X \) as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope. In addition to the material in the extended abstract Kaibel and Weltge (2014) we include omitted proofs, supporting figures, discussions about properties of coefficients in such formulations, and facts about the complexity of formulations in more general settings.


Integer programming Relaxations Auxiliary variables TSP 

Mathematics Subject Classification




We would like to thank Gennadiy Averkov for valuable comments on this work.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Otto-von-Guericke-Universität MagdeburgMagdeburgGermany

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