On Facets of Knapsack Equality Polytopes

  • E. K. Lee


The 0/1 knapsack equality polytope is, by definition, the convex hull of 0/1 solutions of a single linear equation. A special form of this polytope, where the defining linear equation has nonnegative integer coefficients and the number of variables having coefficient one exceeds the right-hand side, is considered. Equality constraints of this form arose in a real-world application of integer programming to a truck dispatching scheduling problem. Families of facet defining inequalities for this polytope are identified, and complete linear inequality representations are obtained for some classes of polytopes.

Knapsack polytopes integer programming polyhedral theory branch-and-cut systems 


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  1. 1.
    BALAS, E., Facets of the Knapsack Polytope, Mathematical Programming, Vol. 8, pp. 146–164, 1975.Google Scholar
  2. 2.
    BALAS, E., and ZEMEL, E., Facets of the Knapsack Polytope from Minimal Covers, SIAM Journal on Applied Mathematics, Vol. 34, pp. 119–148, 1978.Google Scholar
  3. 3.
    HAMMER, P. L., JOHNSON, E. L., and PELED, U. N., Facets of Regular 0–1 Polytopes, Mathematical Programming, Vol. 8, pp. 179–206, 1975.Google Scholar
  4. 4.
    NEMHAUSER, G. L., and WOLSEY, L. A., Integer and Combinatorial Optimization, Wiley, New York, New York, 1988.Google Scholar
  5. 5.
    WOLSEY, L. A., Faces for a Linear Inequality in 0–1 Variables, Mathematical Programming, Vol. 8, pp. 165–178, 1975.Google Scholar
  6. 6.
    ZEMEL, E., Easily Computable Facets of the Knapsack Polytope, Mathematics of Operations Research, Vol. 14, pp. 760–764, 1989.Google Scholar
  7. 7.
    BIXBY, R. E., and LEE, E. K., Solving a Truck Dispatching Scheduling Problem Using Branch-and-Cut, 1994, Operations Research (to appear).Google Scholar
  8. 8.
    LEE, E. K., Solving Structured 0/1 Integer Programming Problems Arising from Truck Dispatching Scheduling Problems, PhD Thesis, Rice University, 1993.Google Scholar
  9. 9.
    LEE, E. K., Facets of Special Knapsack Equality Polytopes, Report 38, Computational and Applied Mathematics, Rice University, 1993.Google Scholar
  10. 10.
    HAMMER, P. L., and PELED, U. N., Computing Low-Capacity 0–1 Knapsack Polytopes, Zeitschrift für Operations Research, Vol. 26, pp. 243–249, 1982.Google Scholar
  11. 11.
    PELED, U. N., Properties of Facets of Binary Polytopes, Annals of Discrete Mathematics, Vol. 1, pp. 435–456, 1977.Google Scholar
  12. 12.
    WEISMANTEL, R., On the 0/1 Knapsack Polytope, Research Report 1, Konrad-Zuse-Zentrum für Informationstechnik, Berlin, Germany, 1994.Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • E. K. Lee
    • 1
  1. 1.Department of Industrial Engineering and Operations ResearchColumbia UniversityNew York

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