Journal of Combinatorial Optimization

, Volume 31, Issue 3, pp 1111–1129 | Cite as

Implicit cover inequalities

  • Agostinho AgraEmail author
  • Cristina Requejo
  • Eulália Santos


In this paper we consider combinatorial optimization problems whose feasible sets are simultaneously restricted by a binary knapsack constraint and a cardinality constraint imposing the exact number of selected variables. In particular, such sets arise when the feasible set corresponds to the bases of a matroid with a side knapsack constraint, for instance the weighted spanning tree problem and the multiple choice knapsack problem. We introduce the family of implicit cover inequalities which generalize the well-known cover inequalities for such feasible sets and discuss the lifting of the implicit cover inequalities. A computational study for the weighted spanning tree problem is reported.


Cover inequalities Weighted minimal spanning tree problem Lifting Matroidal knapsack 



Research partially funded by CIDMA (Centro de Investigação e Desenvolvimento em Matemática e Aplicações) through the FCT (Fundação para a Ciência e a Tecnologia) within project PEst-OE/MAT/UI4106/2014, and by FCT through program COMPETE: FCOMP-01-0124-FEDER-041898 within project EXPL/MAT-NAN/1761/2013.


  1. Aggarwal V, Aneja YP, Nair KPK (1982) Minimal spanning tree subject to a side constraint. Comput Oper Res 9:287–296CrossRefGoogle Scholar
  2. Agra A, Cerveira A, Requejo C, Santos E (2011) On the weight-constrained minimum spanning tree problem. In: Proceedings of the International Network Optimization Conference, volume 6701 of Lecture Notes in Computer Science, pp 156–161Google Scholar
  3. Amado L, Bárcia P (1996) New polynomial bounds for matroidal knapsacks. Eur J Oper Res 95:201–210CrossRefzbMATHGoogle Scholar
  4. Balas E, Jeroslow R (1972) Canonical cuts on the unit hypercube. SIAM J Appl Math 23:61–79MathSciNetCrossRefzbMATHGoogle Scholar
  5. Balas E (1975) Facets of the knapsack polytope. Math Programm 8:146–164MathSciNetCrossRefzbMATHGoogle Scholar
  6. Balas E, Zemel E (1978) Facets of the knapsack polytope from minimal covers. SIAM J Appl Math 34:119–148MathSciNetCrossRefzbMATHGoogle Scholar
  7. Camerini PM, Vercellis C (1984) The matroidal knapsack: a class of (often) well-solved problems. Oper Res Lett 3:157–162MathSciNetCrossRefzbMATHGoogle Scholar
  8. Crowder H, Johnson EL, Padberg MW (1983) Solving large-scale zero-one linear programming problems. Oper Res 31:803–835CrossRefzbMATHGoogle Scholar
  9. Gu Z, Nemhauser GL, Savelsbergh MWP (1998) Lifted cover inequalities for 0–1 integer programs: Computation. INFORMS J Comput 10:427–437MathSciNetCrossRefGoogle Scholar
  10. Gu Z, Nemhauser GL, Savelsbergh MWP (2000) Sequence independent lifting in mixed integer programming. J Combin Optim 10:109–129MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hammer PL, Johnson EL, Peled UN (1975) Facets of regular 0–1 polytopes. Math Programm 8:179–206MathSciNetCrossRefzbMATHGoogle Scholar
  12. Hassin R, Levin A (2004) An efficient polynomial time approximation scheme for the constrained minimum spanning tree problem using matroid intersection. SIAM J Comput 33:261–268MathSciNetCrossRefzbMATHGoogle Scholar
  13. Henn ST (2007) Weight-constrained minimum spanning tree problem. Master’s thesis, University of Kaiserslautern, KaiserslauternGoogle Scholar
  14. Hong SP, Chung SJ, Park BH (2004) A fully polynomial bicriteria approximation scheme for the constrained spanning tree problem. Oper Res Lett 32:233–239MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kaparis K, Letchford AN (2008) Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem. Eur J Oper Res 186:91–103MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kaparis K, Letchford AN (2010) Separation algorithms for 0–1 knapsack polytopes. Math Programm 124:69–91MathSciNetCrossRefzbMATHGoogle Scholar
  17. Klabjan D, Nemhauser GL, Tovey C (1998) The complexity of cover inequality separation. Oper Res Lett 23:35–40MathSciNetCrossRefzbMATHGoogle Scholar
  18. Laurent M (1989) A generalization of antiwebs to independence systems and their canonical facets. Math Programm 45:97–108MathSciNetCrossRefzbMATHGoogle Scholar
  19. Martello S, Toth P (1990) Knapsack problems: algorithms and computer implementations. Wiley, ChichesterzbMATHGoogle Scholar
  20. Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  21. Pisinger D (2005) Where are the hard knapsack problems? Comput Oper Res 32:2271–2284MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ravi R, Goemans MX (1996) The constrained minimum spanning tree problem. In: Proceedings of the Scandinavian Workshop on Algorithmic Theory, volume 1097 of Lecture Notes in Computer Science, pp 66–75Google Scholar
  23. Requejo C, Agra A, Cerveira A, Santos E (2010) Formulations for the weight-constrained minimum spanning tree problem. In: Proceedings of the International Conference on Numerical Analysis and Applied Mathematics, volume 1281 of AIP Conference Proceedings, pp 2166–2169Google Scholar
  24. Requejo C, Santos E (2011) Lagrangean based algorithms for the weight-constrained minimum spanning tree problem. In: Proceedings of the VII ALIO/EURO Workshop on Applied Combinatorial Optimization, pp 38–41Google Scholar
  25. van Roy TJ, Wolsey LA (1987) Solving mixed integer programming problems using automatic reformulation. Oper Res 35:45–57MathSciNetCrossRefzbMATHGoogle Scholar
  26. Shogan A (1983) Constructing a minimal-cost spanning tree subject to resource constraints and flow requirements. Networks 13:169–190MathSciNetCrossRefGoogle Scholar
  27. Wolsey LA (1975) Facets for a linear inequality in 0–1 variables. Math Programm 8:165–178MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yamada T, Watanabe K, Kataoka S (2005) Algorithms to solve the knapsack constrained maximum spanning tree problem. Int J Comput Math 82:23–34MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zemel E (1989) Easily computable facets of the knapsack polytope. Math Oper Res 14:760–765MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Agostinho Agra
    • 1
    Email author
  • Cristina Requejo
    • 1
  • Eulália Santos
    • 2
  1. 1.CIDMA and Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.CIDMAISLA-Higher Institute of LeiriaLeiriaPortugal

Personalised recommendations