Abstract
We consider multi-constrained knapsack problems where the sets of elements to be selected are subject to combinatorial constraints of matroidal nature. For this important class of NP-hard combinatorial optimization problems we prove that Lagrangean relaxation techniques not only provide good bounds to the value of the optimum, but also yield approximate solutions, which are asymptotically optimal under mild probabilistic assumptions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.V. Aho, J.E. Hopcroft and J.D. Ullman,Data Structures and Algorithms (Addison-Wesley, Reading, MA, 1983).
R.D. Armstrong, D.S. Kung, P. Sinha and A.A. Zoltners, “A computational study of a multiple-choice knapsack algorithm,”ACM Transactions on Mathematical Software 9 (1983) 184–198.
J.F. Balintfy, G.T. Ross, P. Sinha and A.A. Zoltners, “A mathematical programming system for preference-maximized nonselective menu planning and scheduling,”Mathematical Programming 15 (1978) 63–76.
D.P. Bertzekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, Cambridge, 1982).
M. Bonatti, P.M. Camerini, L. Fratta, G. Gallassi and F. Maffioli, “A dynamic planning method for telecommunication networks and its performance evaluation for district trunk networks,”Proceedings of 10th International Teletraffic Congress (Montereal, Canada, 1983).
P.M. Camerini and F. Maffioli, “Capacity assignment in PS networks: a hierarchical planning approach,”Proceedings of 11th International Teletraffic Congress (Kyoto, Japan, 1985).
P.M. Camerini and C. Vercellis, “The matroidal knapsack: a class of (often) well-solvable problems,”Operations Research Letters 3 (1984) 157–162.
M.A.H. Dempster, M.L. Fisher, L. Jansen, B.J. Lageweg, J.K. Lenstra and A.H.G. Rinnooy Kan “Analysis of heuristics for stochastic programming results for hierarchical scheduling problems,”Mathematics of Operations Research 8 (1983) 525–537.
R.S. Garfinkel and G.L. Nemhauser,Integer Programming (Wiley, New York, 1972).
A.M. Geoffrion, “Lagrangean relaxation for integer programming,”Mathematical Programming Study 2 (1974) 82–114.
M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees,”Operations Research 18 (1970) 1138–1162.
P. Kolesar, “Assignment of optimal redundancy in systems subject to failure”, Operations Research Group Technical Report, Columbia University (New York, 1966).
E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976).
J. Lorie and L. Savage, “Three problems in capital rationing,”Journal of Business 38 (1955) 229–239.
R.T. Rockafellar, “Generalized subgradients in mathematical programming,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming The State of the Art (Springer, Berlin, 1983) pp. 368–390.
P. Sinha and A.A. Zoltners, “Integer programming models for sales resource allocation,”Management Science 26 (1980) 242–260.
R. Trevisi, “Problemi di ottimizzazione di strutture ad albero con vincoli di affidabilità e di costo: teoria ed applicazioni,” Thesis, Department of Information Science, University of Milano (Milan, 1985).
D.J.A. Welsh,Matroid Theory (Academic Press, London, 1976).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Camerini, P.M., Maffioli, F. & Vercellis, C. Multi-constrained matroidal knapsack problems. Mathematical Programming 45, 211–231 (1989). https://doi.org/10.1007/BF01589104
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01589104