Abstract
The Integer Knapsack Problem with Set-up Weights (IKPSW) is a generalization of the classical Integer Knapsack Problem (IKP), where each item type has a set-up weight that is added to the knapsack if any copies of the item type are in the knapsack solution. The k-item IKPSW (kIKPSW) is also considered, where a cardinality constraint imposes a value k on the total number of items in the knapsack solution. IKPSW and kIKPSW have applications in the area of aviation security. This paper provides dynamic programming algorithms for each problem that produce optimal solutions in pseudo-polynomial time. Moreover, four heuristics are presented that provide approximate solutions to IKPSW and kIKPSW. For each problem, a Greedy heuristic is presented that produces solutions within a factor of 1/2 of the optimal solution value, and a fully polynomial time approximation scheme (FPTAS) is presented that produces solutions within a factor of ε of the optimal solution value. The FPTAS for IKPSW has time and space requirements of O(nlog n+n/ε 2+1/ε 3) and O(1/ε 2), respectively, and the FPTAS for kIKPSW has time and space requirements of O(kn 2/ε 3) and O(k/ε 2), respectively.
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References
Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)
Bruno, J., Downey, P.: Complexity of task scheduling with deadlines, set-up times and changeover costs. SIAM J. Comput. 7(4), 393–404 (1978)
Caprara, A., Kellerer, H., Pferschy, U., Pisinger, D.: Approximation algorithms for knapsack problems with cardinality constraints. Eur. J. Oper. Res. 123(2), 333–345 (2000)
Chajakis, E.D., Guignard, M.: Exact algorithms for the setup knapsack problem. INFOR 32(3), 124–142 (1994)
Ibarra, O.H., Kim, C.E.: Approximation algorithms for certain scheduling problems. Math. Oper. Res. 3(3), 197–204 (1978)
Johnson, D.S., Niemi, K.A.: On knapsacks, partitions, and a new dynamic programming technique for trees. Math. Oper. Res. 8(1), 1–14 (1983)
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Heidelberg (2004)
Lawler, E.L., Fast approximation algorithms for knapsack problems. Math. Oper. Res. 4(4), 339–356 (1979)
Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Implementations. Wiley, New York (1990)
McLay, L.A., Jacobson, S.H.: Integer knapsack problems with set-up weights. Technical report, University of Illinois, Urbana (2005)
McLay, L.A., Jacobson, S.H., Kobza, J.E.: A multilevel passenger screening problem for aviation security. Nav. Res. Log. 53(3), 183–197 (2006)
Park, K., Park, S.: Lifting cover inequalities for the precedence-constrained knapsack problem. Discret. Appl. Math. 72, 219–241 (1997)
Pferschy, U., Pisinger, D., Woeginger, G.J.: Simple but efficient approaches for the collapsing knapsack problem. Discret. Appl. Math. 77, 271–280 (1997)
Pisinger, D.: A fast algorithm for strongly correlated knapsack problems. Discret. Appl. Math. 89, 197–212 (1998)
Süral, H., van Wassenhove, L.N., Potts, C.N.: The bounded knapsack problem with setups. Technical report, INSEAD, Centre for the Management of Environmental Resources, Report 97/71/TM, France (1997)
Yang, W.H., Liao, C.J.: Survey of scheduling research involving setup times. Int. J. Systems Sci. 30(2), 143–155 (1999)
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McLay, L.A., Jacobson, S.H. Integer knapsack problems with set-up weights. Comput Optim Appl 37, 35–47 (2007). https://doi.org/10.1007/s10589-007-9020-5
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DOI: https://doi.org/10.1007/s10589-007-9020-5