Average Case Analysis of the Set Packing Problem

• Autorzy: Krzysztof Szkatuła
• Języki publikacji: EN

Abstrakty

EN

The paper deals with the well known set packing problem and its special case, when the number of subsets is maximized. It is assumed that some of the problem coefficients are realizations of mutually independent random variables. Average case (i.e. asymptotical probabilistic) properties of selected problem characteristics are investigated for the variety of possible instances of the problem. The important results of the paper are: Behavior of the optimal solution values of the set packing problem is presented for the special asymptotic case, where mutual asymptotical relation between m (number of elements of the packed set) and n (number of sets provided) is playing an essential role. Probability of reaching feasible solution is reasonably high (i.e. > 2/e,2/e ≈ 0.736); moreover, it may be set arbitrarily close to 1 (e.g. 0.999), although the deterioration in the quality of approximation of the behavior of the optimal solution values may be substantial.Some relations between the general case of the set packing problem and its maximization for the special case are investigated. (original abstract)

Słowa kluczowe

EN

Operations research; Mathematical optimization; Mathematical programming

PL

Badania operacyjne; Optymalizacja matematyczna; Programowanie matematyczne

• Czasopismo: Control and Cybernetics
• Rocznik: 2014
• Tom: 43
• Numer: nr 4
• Strony: 557--575

Twórcy

autor

Krzysztof Szkatuła

• Systems Research Institute, Polish Academy of Sciences

Bibliografia

• Ausiello, G., D'Atri, A.,, Protasi, M. (1980) Structure preserving reductions among convex optimization problems. J. Comput. System Sci., 21, 136-153.
• Averbakh, I. (1994) Probabilistic properties of the dual structure of the multidimensional knapsack problem and fast statistically effcient algorithms. Mathematical Programming, 65, 311-330.
• Garey, M., Johnson, D. (1972) Computers and Intractability: A Guide to the Theory of NP Completeness. Freeman, San Francisco.
• Karp, R. (1979) Reducibility among combinatorial problems In: R. Miller and J. Thatcher, eds., Complexity of Computer Computations. Plenum Press, New York.
• Martello, S., Toth, P. (1990) Knapsack Problems: Algorithms and Computer Implementations. Wiley & Sons.
• Meanti, M., Kan, A.R., Stougie, L., Vercellis, C. (1990) A probabilistic analysis of the multiknapsack value function. Mathematical Programming, 46, 237-247.
• Nemhauser, G., Wolsey, L. (1988) Integer and Combinatorial Optimization. John Wiley & Sons Inc., New York.
• Szkatula, K. (1994) On the growth of multi-constraint random knapsacks with various right hand sides of the constraints. European Journal of Operational Reserch, 73, 199-204.
• Szkatula, K. (1997) The growth of multi-constraint random knapsacks with large right-hand sides of the constraints. Operations Research Letters, 21, 25-30.
• Vercellis, C. (1986) A probabilistic analysis of the set packing problem. In: Stochastic Programming. Springer Verlag, 272-285.