Mathematical Programming

, Volume 99, Issue 2, pp 223–239 | Cite as

On unions and dominants of polytopes

  • Egon Balas
  • Alexander Bockmayr
  • Nicolai Pisaruk
  • Laurence Wolsey


A well-known result on unions of polyhedra in the same space gives an extended formulation in a higher-dimensional space whose projection is the convex hull of the union. Here in contrast we study the unions of polytopes in different spaces, giving a complete description of the convex hull without additional variables. When the underlying polytopes are monotone, the facets are described explicitly, generalizing results of Hong and Hooker on cardinality rules, and an efficient separation algorithm is proposed. These results are based on an explicit representation of the dominant of a polytope, and an algorithm for the separation problem for the dominant. For non-monotone polytopes, both the dominant and the union are characterized. We also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of Conforti and Laurent.


unions of polytopes cardinality constraints separation convex hulls dominant matroids polymatroids 


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  1. 1.
    Balas, E.: Disjunctive programming: Properties of the convex hull of feasible points. Invited paper with foreword by G. Cornuéjols and W.R. Pulleyblank. Disc. Appl. Math. 89, 1–44 (1998)MATHGoogle Scholar
  2. 2.
    Balas, E.: Logical Constraints as Cardinality Rules: Tight Representations. #MSRR-628, Carnegie Mellon University, 2000Google Scholar
  3. 3.
    Balas, E., Fischetti, M.: On the monotonization of polyhedra. Math. Program. 78, 59–84 (1997)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Boyd, S.C., Pulleyblank, W.R.: Facet generating techniques, Department of Combinatorics and Optimization. University of Waterloo, 1991Google Scholar
  5. 5.
    Conforti, M., Laurent, M.: On the facial structure of independence system polyhedra. Math. Oper. Res. 13, 543–555 (1988)MathSciNetMATHGoogle Scholar
  6. 6.
    Cunningham, W.H.: Testing membership in matroid polyhedra. J. Combin. Theory B 36, 161–188 (1984)MathSciNetMATHGoogle Scholar
  7. 7.
    Cunningham, W.H.: On submodular function minimization. Combinatorica 5, 185–192 (1985)MathSciNetMATHGoogle Scholar
  8. 8.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J., eds. Combinatorial Structures and their applications, Gordon and Breach, New York, 1970, pp. 69–87Google Scholar
  9. 9.
    Edmonds, J.: Matroids and the Greedy Algorithm. Math. Program. 1, 127–136 (1971)MATHGoogle Scholar
  10. 10.
    Edmonds, J., Fulkerson, D.R.: Transversals and matroid partition. J. Res. Nat. Bur. Standards 69B, 147–153 (1965)MATHGoogle Scholar
  11. 11.
    Fulkerson, D.R.: Blocking and Antiblocking Pairs of Polyhedra. Math. Program. 1, 168–194 (1971)MATHGoogle Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer Verlag 1988Google Scholar
  13. 13.
    Hong, Y., Hooker, J.N.: Tight representation of logical constraints as cardinality rules. Math. Program. 85, 363–377 (1999)CrossRefGoogle Scholar
  14. 14.
    Iwata, S., Fleisher, L., Fujishige, S.: A combinatorial, strongly polynomial-time algorithm for minimizing submodular functions. J. ACM 48, 761–777 (2001)CrossRefGoogle Scholar
  15. 15.
    Padberg, M.W.: (1,k)-Configurations and facets for packing problems. Math. Program. 18, 94–99 (1980)MathSciNetMATHGoogle Scholar
  16. 16.
    Rockafellar, T.: Convex analysis. Princeton University Press, 1970Google Scholar
  17. 17.
    Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Combinatorial Theory, Ser. B 80, 346–355 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Egon Balas
    • 1
  • Alexander Bockmayr
    • 2
  • Nicolai Pisaruk
    • 2
  • Laurence Wolsey
    • 3
  1. 1.Graduate School of Industrial AdministrationCarnegie-Mellon UniversityPittsburgh PAUSA
  2. 2.Université Henri PoincaréLORIAVandoeuvre-lès-NancyFrance
  3. 3.CORE and INMAUniversité Catholique de LouvainLouvain-la-NeuveBelgium

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