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Mathematical Programming

, Volume 135, Issue 1–2, pp 323–335 | Cite as

Vectors in a box

  • Kevin BuchinEmail author
  • Jiří Matoušek
  • Robin A. Moser
  • Dömötör Pálvölgyi
Open Access
Full Length Paper Series A

Abstract

For an integer d ≥ 1, let τ(d) be the smallest integer with the following property: if v 1, v 2, . . . , v t is a sequence of t ≥ 2 vectors in [−1, 1] d with \({{\bf v}_1+{\bf v}_2+\cdots+{\bf v}_t \in [-1,1]^d}\) , then there is a set \({S\subseteq \{1,2,\ldots,t\}}\) of indices, 2 ≤ |S| ≤ τ(d), such that \({\sum_{i \in S}{\bf v}_i \in [-1,1]^d}\) . The quantity τ(d) was introduced by Dash, Fukasawa, and Günlük, who showed that τ(2) = 2, τ(3) = 4, and τ(d) = Ω(2 d ), and asked whether τ(d) is finite for all d. Using the Steinitz lemma, in a quantitative version due to Grinberg and Sevastyanov, we prove an upper bound of τ(d) ≤ d d+o(d), and based on a construction of Alon and Vũ, whose main idea goes back to Håstad, we obtain a lower bound of τ(d) ≥ d d/2-o(d). These results contribute to understanding the master equality polyhedron with multiple rows defined by Dash et al. which is a “universal” polyhedron encoding valid cutting planes for integer programs (this line of research was started by Gomory in the late 1960s). In particular, the upper bound on τ(d) implies a pseudo-polynomial running time for an algorithm of Dash et al. for integer programming with a fixed number of constraints. The algorithm consists in solving a linear program, and it provides an alternative to a 1981 dynamic programming algorithm of Papadimitriou.

Keywords

Steinitz lemma Integer programming Master equality polyhedron 

Mathematics Subject Classification (2000)

52B05 90C10 

Notes

Acknowledgments

This research was partially done at the Gremo Workshop on Open Problems 2009, and the support of the ETH Zürich is gratefully acknowledged. We would like to thank Tibor Szabó for raising the problem at the GWOP’09 workshop, Sanjeeb Dash for prompt answers to our questions, and Patrick Traxler for useful discussions. The first author was supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 639.022.707. The European Union and the European Social Fund have provided financial support for the fourth author to the project under the grant agreement no. TÁMOP 4.2.1./B-09/1/KMR-2010-0003. The research of the third author was partially done during an internship with Microsoft Research, Redmond, Washington, USA.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Kevin Buchin
    • 1
    Email author
  • Jiří Matoušek
    • 2
    • 3
  • Robin A. Moser
    • 3
  • Dömötör Pálvölgyi
    • 4
  1. 1.Department of Mathematics and Computer ScienceTechnical University of EindhovenEindhovenThe Netherlands
  2. 2.Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic
  3. 3.Institute of Theoretical Computer ScienceETH ZurichZurichSwitzerland
  4. 4.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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