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Modeling and Optimization: Theory and Applications pp 77–94Cite as

A few strong knapsack facets

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 147)

Abstract

We perform a shooting experiment for the knapsack facets and observe that 1∕k-facets are strong for small k; in particular, k dividing 6 or 8. We also observe spikes of the size of 1∕k-facets when k = n or when k + 1 divides n + 1. We discuss the strength of the 1∕n-facets introduced by Aráoz et al. (Math Program 96:377–408, 2003) and the knapsack facets given by Gomory’s homomorphic lifting.A general integer knapsack problem is a knapsack subproblem where a portion, often a significant majority, of the variables are missing from the master knapsack problem. The number of projections of 1∕k-facets on a knapsack subproblem of l variables is \(O(l^{\lceil k/2\rceil })\), note that this is independent of the size of the master problem. Since 1∕k-facets are strong for small k, we define the 1∕k-inequalities which include the 1∕d-facets with d dividing k and fix k to be a small constant such as k = 6 or k = 8. We develop an efficient way of enumerating violated valid 1∕k-inequalities. For each violated 1∕k-inequality, we determine its validity by solving a small integer programming problem, the size of which depends only on k.

Keywords

  • Knapsack problem
  • Master knapsack polytope
  • Facets
  • Shooting experiment
  • Cutting planes

Mathematics subject classification (2010):

  • 90C10 Integer Programming

This chapter was prepared for publication in MOPTA 2014 Proceedings.

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  • DOI: 10.1007/978-3-319-23699-5_4
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Author information

Authors and Affiliations

  1. Kellogg Graduate School of Management, Northwestern University, Leverone Hall, Evanston, IL, 60208, USA

    Sunil Chopra

  2. Department of Engineering, Robert Morris University, Moon Twp, PA, 15108, USA

    Sangho Shim

  3. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    Daniel E. Steffy

Authors
  1. Sunil Chopra
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  2. Sangho Shim
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  3. Daniel E. Steffy
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Corresponding author

Correspondence to Sangho Shim .

Editor information

Editors and Affiliations

  1. Dept. of Industrial & Systems Eng., Lehigh University, Bethlehem, Pennsylvania, USA

    Boris Defourny

  2. Dept. of Industrial & Systems Eng., Lehigh University, Bethlehem, Pennsylvania, USA

    Tamás Terlaky

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Chopra, S., Shim, S., Steffy, D.E. (2015). A few strong knapsack facets. In: Defourny, B., Terlaky, T. (eds) Modeling and Optimization: Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 147. Springer, Cham. https://doi.org/10.1007/978-3-319-23699-5_4

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