In this paper we propose new lower and upper bounds for the max-min 0-1 knapsack problem, employing a mixture of two relaxations. In addition, in order to expose whether the bounds are practical or not, we implement a...
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In this paper we propose new lower and upper bounds for the max-min 0-1 knapsack problem, employing a mixture of two relaxations. In addition, in order to expose whether the bounds are practical or not, we implement a method incorporating the bounds to achieve an optimal solution of the problem.
The number partitioning problem consists of partitioning a sequence of positive numbers {a(1),a(2),...,a(N)} into two disjoint sets, A and B, such that the absolute value of the difference of the sums of ai over the t...
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The number partitioning problem consists of partitioning a sequence of positive numbers {a(1),a(2),...,a(N)} into two disjoint sets, A and B, such that the absolute value of the difference of the sums of ai over the two sets is minimized. We use statistical mechanics tools to study analytically the linear programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of A and B and shaw that this difference is not self-averaging. (C) 1999 Elsevier Science B.V. All rights reserved.
New relaxations are developed in this paper for problems of optimal packing of small (rectangular-shaped) pieces within one or several larger containers. Based on these relaxations tighter bounds for the Container Loa...
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In this paper the solution of two-stage guillotine cutting stock problems is considered. Especially such problems are under investigation where the sizes of the order demands differ in a large range. We propose a new ...
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In this paper the solution of two-stage guillotine cutting stock problems is considered. Especially such problems are under investigation where the sizes of the order demands differ in a large range. We propose a new approach dealing with such situations and compare it with the classical Gilmore-Gomory approach. We report results of extensive numerical experiments which show the advantages of the new approach.
We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinearprogramming problems. Our techniques find the optimal solution value and the optimal dual...
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We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinearprogramming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, the k-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems, K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.
A class of generalized greedy algorithms is proposed for the solution of the {0, 1} multi-knapsack problem. Items are selected according to decreasing ratios of their profit and a weighted sum of their requirement coe...
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A class of generalized greedy algorithms is proposed for the solution of the {0, 1} multi-knapsack problem. Items are selected according to decreasing ratios of their profit and a weighted sum of their requirement coefficients. The solution obtained depends on the choice of the weights. A geometrical representation of the method is given and the relation to the dual of the linear programming relaxation of multi-knapsack is exploited. We investigate the complexity of computing a set of weights that gives the maximum greedy solution value. Finally, the heuristics are subjected to both a worst-case and a probabilistic performance analysis.
We analyze probabilistically the classical Held-Karp lower bound derived from the 1-tree relaxation for the Euclidean traveling salesman problem (ETSP). We prove that, if n points are identically and independently dis...
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We analyze probabilistically the classical Held-Karp lower bound derived from the 1-tree relaxation for the Euclidean traveling salesman problem (ETSP). We prove that, if n points are identically and independently distributed according to a distribution with bounded support and absolutely continuous part f(x) dx over the d-cube, the Held-Karp lower bound on these n points is almost surely asymptotic to beta-HK(d)n(d-1)/d integral f(x)(d-1)/d dx, where beta-HK(d) is a constant independent of n. The result suggests a probabilistic explanation of the observation that the lower bound is very close to the length of the optimal tour in practice, since the ETSP is almost surely asymptotic to beta-TSP(d)n(d-1)/d integral f(x)(d-1)/d dx. The techniques we use exploit the polyhedral description of the Held-Karp lower bound and the theory of subadditive Euclidean functionals.
A unified approach and a summary of the most important results concerned with exact methods for solving the (binary) knapsack problem and its generalizations are given. We stress the importance of dual methods for sol...
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A unified approach and a summary of the most important results concerned with exact methods for solving the (binary) knapsack problem and its generalizations are given. We stress the importance of dual methods for solving linear programming relaxations of the considered problems. Two ways of generalization of the knapsack problem are described. If the special ordered sets are added, then the multiple-choice knapsack problem is obtained. If the constraints have the nested structure, then we get the nested knapsack problem. Also the multiple-choice nested knapsack problem is discussed.
We study the gap between the value of the integer solution to the set covering problem and the value of the fractional solution that solves the linear programming relaxation of this problem.
We study the gap between the value of the integer solution to the set covering problem and the value of the fractional solution that solves the linear programming relaxation of this problem.
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