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Optimization algorithms for the disjunctively constrained knapsack problem

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Abstract

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.

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References

  • Akeb H, Hifi M, Mounir MEOA (2011) Local branching-based algorithms for the disjunctively constrained knapsack problem. Comput Ind Eng 60(4):811–820

    Article  Google Scholar 

  • Atamtürk A, Narayanan V (2009) The submodular knapsack polytope. Discrete Optim 6(4):333–344

  • Balas E (1975) Facets of the knapsack polytope. Math Program 8(1):146–164

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Zemel E (1978) Facets of the knapsack polytope from minimal covers. SIAM J Appl Math 34(1):119–148

    Article  MathSciNet  MATH  Google Scholar 

  • 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, Italy

  • Boyd EA (1993) Polyhedral results for the precedence-constrained knapsack problem. Discrete Appl Math 41(3):185–201

    Article  MathSciNet  MATH  Google Scholar 

  • Crowder H, Johnson EL, Padberg M (1983) Solving large-scale zero-one linear programming problems. Oper Res 31(5):803–834

    Article  MATH  Google Scholar 

  • 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):451–462

    Article  MathSciNet  MATH  Google Scholar 

  • de Farias Jr IR, Nemhauser GL (2003) A polyhedral study of the cardinality constrained knapsack problem. Math Program 96(3):439–467

    Article  MathSciNet  MATH  Google Scholar 

  • 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):252–264

    Article  MathSciNet  MATH  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of np-completeness. Freeman, San Francisco

    MATH  Google Scholar 

  • Grötschel M, Lovász L, Schrijver A (2012) Geometric algorithms and combinatorial optimization. Springer, Berlin

    MATH  Google Scholar 

  • Gu Z, Nemhauser GL, Savelsbergh MW (1998) Lifted cover inequalities for 0–1 integer programs: computation. INFORMS J Comput 10(4):427–437

    Article  MathSciNet  Google Scholar 

  • Hammer PL, Johnson EL, Peled UN (1975) Facet of regular 0–1 polytopes. Math Program 8(1):179–206

    Article  MathSciNet  MATH  Google Scholar 

  • Hanafi S, Glover F (2007) Exploiting nested inequalities and surrogate constraints. Eur J Oper Res 179(1):50–63

    Article  MathSciNet  MATH  Google Scholar 

  • Hifi M, Michrafy M (2007) Reduction strategies and exact algorithms for the disjunctively constrained knapsack problem. Comput Oper Res 34(9):2657–2673

    Article  MATH  Google Scholar 

  • 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 279–284

  • 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 1547–1551

  • 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 1–6

  • Hifi M, Saleh S, Wu L, Chen J (2015) A hybrid guided neighborhood search for the disjunctively constrained knapsack problem. Cogent Eng 2(1):1068,969

  • Kaparis K, Letchford AN (2008) Local and global lifted cover inequalities for the 0–1 multidimensional knapsack problem. Eur J Oper Res 186(1):91–103

    Article  MathSciNet  MATH  Google Scholar 

  • Kaparis K, Letchford AN (2010a) Cover inequalities. Wiley Encyclopedia of Operations Research and Management Science

  • Kaparis K, Letchford AN (2010b) Separation algorithms for 0–1 knapsack polytopes. Math Program 124(1–2):69–91

    Article  MathSciNet  MATH  Google Scholar 

  • Klabjan D, Nemhauser GL, Tovey C (1998) The complexity of cover inequality separation. Oper Res Lett 23(1):35–40

    Article  MathSciNet  MATH  Google Scholar 

  • Mahjoub AR (2010) Polyhedral approaches. In: Paschos V (ed) Concepts of combinatorial optimization. ISTE-Wiely, pp 261–324

  • Martello S, Pisinger D, Toth P (1997) Dynamic programming and tight bounds for the 0–1 knapsack problem. Københavns Universitet, Datalogisk Institut

    MATH  Google Scholar 

  • Martello S, Toth P (1990) Knapsack problems: algorithms and computer implementations. Wiley, New York

    MATH  Google Scholar 

  • Nemhauser G, Sigismondi G (1992) A strong cutting plane/branch-and-bound algorithm for node packing. J Oper Res Soc 43(5):443–457

    Article  MATH  Google Scholar 

  • Nemhauser GL, Trotter LE Jr (1974) Properties of vertex packing and independence system polyhedra. Math Program 6(1):48–61

    Article  MathSciNet  MATH  Google Scholar 

  • Pferschy U, Schauer J (2009) The knapsack problem with conflict graphs. J Graph Algorithms Appl 13(2):233–249

    Article  MathSciNet  MATH  Google Scholar 

  • Pisinger D (1999) Core problems in knapsack algorithms. Oper Res 47(4):570–575

    Article  MathSciNet  MATH  Google Scholar 

  • Sadykov R, Vanderbeck F (2013) Bin packing with conflicts: a generic branch-and-price algorithm. INFORMS J Comput 25(2):244–255

    Article  MathSciNet  Google Scholar 

  • Schrijver A (2002) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin

    MATH  Google Scholar 

  • Senisuka A, You B, Yamada T (2005) Reduction and exact algorithms for the disjunctively constrained knapsack problem. In: International symposium, operational research Bremen

  • Van Roy TJ, Wolsey LA (1987) Solving mixed integer programming problems using automatic reformulation. Oper Res 35(1):45–57

    Article  MathSciNet  MATH  Google Scholar 

  • Weismantel R (1997) On the 0/1 knapsack polytope. Math Program 77(3):49–68

    Article  MathSciNet  MATH  Google Scholar 

  • Wolsey LA (1975) Faces for a linear inequality in 0–1 variables. Math Program 8(1):165–178

    Article  MathSciNet  MATH  Google Scholar 

  • Wolsey LA, Nemhauser GL (1999) Integer and combinatorial optimization. Wiley-Interscience, New York

    MATH  Google Scholar 

  • Yamada T, Kataoka S, Watanabe K (2002) Heuristic and exact algorithms for the disjunctively constrained knapsack problem. Inform Proces Soc Jpn J 43(9)

  • Zemel E (1989) Easily computable facets of the knapsack polytope. Math Oper Res 14(4):760–764

    Article  MathSciNet  MATH  Google Scholar 

  • 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):277–301

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Correspondence to A. Ridha Mahjoub.

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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.

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

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Communicated by V. Loia.

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Ben Salem, M., Taktak, R., Mahjoub, A.R. et al. Optimization algorithms for the disjunctively constrained knapsack problem. Soft Comput 22, 2025–2043 (2018). https://doi.org/10.1007/s00500-016-2465-7

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