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Local and global lifted cover inequalities for the 0-1 multidimensional knapsack problem. (English) Zbl 1138.90016
Summary: The 0-1 multidimensional knapsack problem (0-1 MKP) is a well-known (and strongly $$\mathcal {NP}$$-hard) combinatorial optimization problem with many applications. Up to now, the majority of upper bounding techniques for the 0-1 MKP have been based on Lagrangian or surrogate relaxation. We show that good upper bounds can be obtained by a cutting plane method based on lifted cover inequalities (LCIs). As well as using traditional LCIs, we use some new ‘global’ LCIs, which take the whole constraint matrix into account.

##### MSC:
 90C10 Integer programming 90C27 Combinatorial optimization
##### Keywords:
integer programming; combinatorial optimization
Full Text:
##### References:
 [1] Atamturk, A., Cover and pack inequalities for (mixed) integer programming, Annals of operations research, 139, 21-38, (2005) · Zbl 1091.90053 [2] Balas, E., Facets of the knapsack polytope, Mathematical programming, 8, 146-164, (1975) · Zbl 0316.90046 [3] Balas, E.; Zemel, E., Facets of the knapsack polytope from minimal covers, SIAM journal on applied mathematics, 34, 119-148, (1978) · Zbl 0385.90083 [4] Bektas, T.; Oguz, O., On separating cover inequalities for the multidimensional knapsack problem, Computers and operations research, 34, 1771-1776, (2007) · Zbl 1159.90460 [5] Boyd, E.A., Generating Fenchel cutting planes for knapsack polyhedra, SIAM journal on optimization, 3, 734-750, (1993) · Zbl 0797.90067 [6] Chvátal, V., Edmonds polytopes and a hierarchy of combinatorial problems, Discrete mathematics, 4, 305-337, (1973) · Zbl 0253.05131 [7] Chu, P.C.; Beasley, J.E., A genetic algorithm for the multidimensional knapsack problem, Journal of heuristics, 4, 63-86, (1998) · Zbl 0913.90218 [8] Crowder, H.; Johnson, E.; Padberg, M., Solving large-scale 0-1 linear programming programs, Operations research, 31, 803-834, (1983) · Zbl 0576.90065 [9] Ferreira, C.E.; Martin, A.; Weismantel, R., Solving multiple knapsack problems by cutting planes, SIAM journal on optimizaton, 6, 858-877, (1996) · Zbl 0856.90082 [10] Fréville, A., The multidimensional 0-1 knapsack problem: an overview, European journal of operational research, 155, 1-21, (2004) · Zbl 1045.90050 [11] Fréville, A.; Hanafi, S., Multidimensional 0-1 knapsack problem: bounds and computational aspects, Annals of operations research, 139, 195-227, (2005) · Zbl 1091.90042 [12] Gabrel, V.; Minoux, M., A scheme for exact separation of extended cover inequalities and application to multidimensional knapsack problems, Operations research letters, 30, 252-264, (2002) · Zbl 1049.90074 [13] Garey, M.R.; Johnson, D.S., Computers and intractability: an introduction to the theory of $$\mathcal{NP}$$-completeness, (1979), W.H. Freeman San Francisco [14] Gu, Z.; Nemhauser, G.L.; Savelsbergh, M.W.P., Cover inequalities for 0-1 linear programs: computation, Informs journal on computing, 10, 427-437, (1998) [15] Gu, Z.; Nemhauser, G.L.; Savelsbergh, M.W.P., Cover inequalities for 0-1 linear programs: complexity, Informs journal on computing, 11, 117-123, (1999) · Zbl 1092.90527 [16] Gu, Z.; Nemhauser, G.L.; Savelsbergh, M.W.P., Sequence independent lifting in mixed integer programming, Journal of combinatorial optimization, 4, 109-129, (2000) · Zbl 0964.90030 [17] Guignard, M.; Kim, S., Lagrangean decomposition: A model yielding strong Lagrangean bounds, Mathematical programming, 39, 215-228, (1987) · Zbl 0638.90074 [18] Klabjan, D.; Nemhauser, G.; Tovey, C., The complexity of cover inequality separation, Operations research letters, 23, 35-40, (1998) · Zbl 0957.90094 [19] Letchford, A.N.; Lodi, A., Strengthening chvátal – gomory cuts and Gomory fractional cuts, Operations research letters, 32, 74-82, (2002) · Zbl 1027.90062 [20] Martin, A.; Weismantel, R., The intersection of knapsack polyhedra and extensions, () · Zbl 0910.90223 [21] Nemhauser, G.L.; Wolsey, L.A., Integer and combinatorial optimization, (1988), Wiley New York · Zbl 0469.90052 [22] Nobili, P.; Sassano, A., A separation routine for the set covering polytope, () · Zbl 0712.90060 [23] Kellerer, H.; Pferschy, U.; Pisinger, D., Knapsack problems, (2004), Springer-Verlag · Zbl 1103.90003 [24] Wolsey, L.A., Faces for linear inequalities in 0-1 variables, Mathematical programming, 8, 165-178, (1975) · Zbl 0314.90063 [25] Wolsey, L.A., Facets and strong valid inequalities for integer programs, Operations research, 24, 367-372, (1976) · Zbl 0339.90036 [26] Zemel, E., Easily computable facets of the knapsack polytope, Mathematics of operations research, 14, 760-765, (1989) · Zbl 0689.90057
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