Soft Computing

, Volume 22, Issue 6, pp 2025–2043 | Cite as

Optimization algorithms for the disjunctively constrained knapsack problem

  • Mariem Ben Salem
  • Raouia Taktak
  • A. Ridha Mahjoub
  • Hanêne Ben-Abdallah
Methodologies and Application


This paper deals with the Knapsack Problem with conflicts, also known as the Disjunctively Constrained Knapsack Problem. The conflicts are represented by a graph whose vertices are the items such that adjacent items cannot be packed in the knapsack simultaneously. We consider a classical formulation for the problem, study the polytope associated with this formulation and investigate the facial aspect of its basic constraints. We then present new families of valid inequalities and describe necessary and sufficient conditions for these inequalities to be facet defining. We also devise separation routines for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem. An extensive computational study is also presented.


Knapsack problem Disjunctive constraints Polytope Facet Separation Branch-and-cut 



We would like to thank the referees for their valuable comments which helped to improve the presentation of the paper.

Compliance with ethical standards

Conflict of interest

Mariem Ben Salem declares that she has no conflict of interest. Dr. Raouia Taktak declares that she has no conflict of interest. Prof. Dr. A. Ridha Mahjoub declares that he has no conflict of interest. Prof. Dr. Hanêne Ben-Abdallah declares that she has no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Akeb H, Hifi M, Mounir MEOA (2011) Local branching-based algorithms for the disjunctively constrained knapsack problem. Comput Ind Eng 60(4):811–820CrossRefGoogle Scholar
  2. Atamtürk A, Narayanan V (2009) The submodular knapsack polytope. Discrete Optim 6(4):333–344Google Scholar
  3. Balas E (1975) Facets of the knapsack polytope. Math Program 8(1):146–164MathSciNetCrossRefzbMATHGoogle Scholar
  4. Balas E, Zemel E (1978) Facets of the knapsack polytope from minimal covers. SIAM J Appl Math 34(1):119–148MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bettinelli A, Cacchiani V, Malaguti E (2014) Bounds and algorithms for the knapsack problem with conflict graph. Tech. rep., Technical Report OR-14-16, DEIS–University of Bologna, Bologna, ItalyGoogle Scholar
  6. Boyd EA (1993) Polyhedral results for the precedence-constrained knapsack problem. Discrete Appl Math 41(3):185–201MathSciNetCrossRefzbMATHGoogle Scholar
  7. Crowder H, Johnson EL, Padberg M (1983) Solving large-scale zero-one linear programming problems. Oper Res 31(5):803–834CrossRefzbMATHGoogle Scholar
  8. Euler R, Jünger M, Reinelt G (1987) Generalizations of cliques, odd cycles and anticycles and their relation to independence system polyhedra. Math Oper Res 12(3):451–462MathSciNetCrossRefzbMATHGoogle Scholar
  9. de Farias Jr IR, Nemhauser GL (2003) A polyhedral study of the cardinality constrained knapsack problem. Math Program 96(3):439–467MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gabrel V, Minoux M (2002) A scheme for exact separation of extended cover inequalities and application to multidimensional knapsack problems. Oper Res Lett 30(4):252–264MathSciNetCrossRefzbMATHGoogle Scholar
  11. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of np-completeness. Freeman, San FranciscozbMATHGoogle Scholar
  12. Grötschel M, Lovász L, Schrijver A (2012) Geometric algorithms and combinatorial optimization. Springer, BerlinzbMATHGoogle Scholar
  13. Gu Z, Nemhauser GL, Savelsbergh MW (1998) Lifted cover inequalities for 0–1 integer programs: computation. INFORMS J Comput 10(4):427–437MathSciNetCrossRefGoogle Scholar
  14. Hammer PL, Johnson EL, Peled UN (1975) Facet of regular 0–1 polytopes. Math Program 8(1):179–206MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hanafi S, Glover F (2007) Exploiting nested inequalities and surrogate constraints. Eur J Oper Res 179(1):50–63MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hifi M, Michrafy M (2007) Reduction strategies and exact algorithms for the disjunctively constrained knapsack problem. Comput Oper Res 34(9):2657–2673CrossRefzbMATHGoogle Scholar
  17. Hifi M, Negre S, Mounir MQA (2009) Local branching-based algorithm for the disjunctively constrained knapsack problem. In: IEEE international conference on computers and industrial engineering, 2009. pp 279–284Google Scholar
  18. Hifi M, Negre S, Saadi T, Saleh S, Wu L (2014) A parallel large neighborhood search-based heuristic for the disjunctively constrained knapsack problem. In: IEEE international processing symposium workshops (IPDPSW) parallel and distributed, pp 1547–1551Google Scholar
  19. Hifi M, Otmani N (2011) A first level scatter search for disjunctively constrained knapsack problems. In: IEEE international conference on communications, computing and control applications (CCCA). pp 1–6Google Scholar
  20. Hifi M, Saleh S, Wu L, Chen J (2015) A hybrid guided neighborhood search for the disjunctively constrained knapsack problem. Cogent Eng 2(1):1068,969Google Scholar
  21. Kaparis K, Letchford AN (2008) Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem. Eur J Oper Res 186(1):91–103MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kaparis K, Letchford AN (2010a) Cover inequalities. Wiley Encyclopedia of Operations Research and Management ScienceGoogle Scholar
  23. Kaparis K, Letchford AN (2010b) Separation algorithms for 0–1 knapsack polytopes. Math Program 124(1–2):69–91MathSciNetCrossRefzbMATHGoogle Scholar
  24. Klabjan D, Nemhauser GL, Tovey C (1998) The complexity of cover inequality separation. Oper Res Lett 23(1):35–40MathSciNetCrossRefzbMATHGoogle Scholar
  25. Mahjoub AR (2010) Polyhedral approaches. In: Paschos V (ed) Concepts of combinatorial optimization. ISTE-Wiely, pp 261–324Google Scholar
  26. Martello S, Pisinger D, Toth P (1997) Dynamic programming and tight bounds for the 0–1 knapsack problem. Københavns Universitet, Datalogisk InstitutzbMATHGoogle Scholar
  27. Martello S, Toth P (1990) Knapsack problems: algorithms and computer implementations. Wiley, New YorkzbMATHGoogle Scholar
  28. Nemhauser G, Sigismondi G (1992) A strong cutting plane/branch-and-bound algorithm for node packing. J Oper Res Soc 43(5):443–457CrossRefzbMATHGoogle Scholar
  29. Nemhauser GL, Trotter LE Jr (1974) Properties of vertex packing and independence system polyhedra. Math Program 6(1):48–61MathSciNetCrossRefzbMATHGoogle Scholar
  30. Pferschy U, Schauer J (2009) The knapsack problem with conflict graphs. J Graph Algorithms Appl 13(2):233–249MathSciNetCrossRefzbMATHGoogle Scholar
  31. Pisinger D (1999) Core problems in knapsack algorithms. Oper Res 47(4):570–575MathSciNetCrossRefzbMATHGoogle Scholar
  32. Sadykov R, Vanderbeck F (2013) Bin packing with conflicts: a generic branch-and-price algorithm. INFORMS J Comput 25(2):244–255MathSciNetCrossRefGoogle Scholar
  33. Schrijver A (2002) Combinatorial optimization: polyhedra and efficiency. Springer, BerlinzbMATHGoogle Scholar
  34. Senisuka A, You B, Yamada T (2005) Reduction and exact algorithms for the disjunctively constrained knapsack problem. In: International symposium, operational research BremenGoogle Scholar
  35. Van Roy TJ, Wolsey LA (1987) Solving mixed integer programming problems using automatic reformulation. Oper Res 35(1):45–57MathSciNetCrossRefzbMATHGoogle Scholar
  36. Weismantel R (1997) On the 0/1 knapsack polytope. Math Program 77(3):49–68MathSciNetCrossRefzbMATHGoogle Scholar
  37. Wolsey LA (1975) Faces for a linear inequality in 0–1 variables. Math Program 8(1):165–178MathSciNetCrossRefzbMATHGoogle Scholar
  38. Wolsey LA, Nemhauser GL (1999) Integer and combinatorial optimization. Wiley-Interscience, New YorkzbMATHGoogle Scholar
  39. Yamada T, Kataoka S, Watanabe K (2002) Heuristic and exact algorithms for the disjunctively constrained knapsack problem. Inform Proces Soc Jpn J 43(9)Google Scholar
  40. Zemel E (1989) Easily computable facets of the knapsack polytope. Math Oper Res 14(4):760–764MathSciNetCrossRefzbMATHGoogle Scholar
  41. Zeng B, Richard JPP (2011) A polyhedral study on 0–1 knapsack problems with disjoint cardinality constraints: facet-defining inequalities by sequential lifting. Discrete Optim 8(2):277–301MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.FSEGS/MIRACL, Université de SfaxSfaxTunisia
  2. 2.ISIMS/CRNS, Université de SfaxSfaxTunisia
  3. 3.Université Paris-Dauphine, PSL Research University, CNRS, LAMSADEParisFrance
  4. 4.King Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations