Gérard Plateau

An algorithm for the solution of the 0–1 knapsack problem

A new implicit enumeration algorithm for the solution of the 0–1 knapsack problem — denoted by FPK 79 — is proposed. The implementation of the associated FORTRAN IV subroutine is then described. Computational results prove the efficiency of this algorithm (practically linear time complexity including the initial arrangement of the data) whose performance is generally better than that of algorithm 37 and thus superior to that of the best known algorithms.ZusammenfassungWir stellen einen neuen Enumerationsalgorithmus — FPK 79 genannt — für die Lösung des 0–1 Knapsack Problems vor. Dann beschreiben wir die zugehörige Fortran IV Subroutine. Die durchgeführten numerischen Versuche zeigen experimentell, daß der Algorithmus einschließlich des Sortierens der Eingangsdaten lineares Zeitverhalten aufweist. Er ist damit leistungsfähiger als der Algorithmus 37 und somit besser als die besten bekannten Algorithmen.

An efficient preprocessing procedure for the multidimensional 0–1 knapsack problem

Abstract The multidimensional 0–1 knapsack problem, defined as a knapsack with multiple resource constraints, is well known to be much more difficult than the single constraint version. This paper deals with the design of an efficient preprocessing procedure for large-scale instances. The algorithm provides sharp lower and upper bounds on the optimal value, and also a tighter equivalent representation by reducing the continuous feasible set and by eliminating constraints and variables. This scheme is shown to be very effective through a lot of computational experiments with test problems of the literature and large-scale randomly generated instances.

RESOLUTION OF THE 0-1 KNAPSACK PROBLEM: COMPARISON OF METHODS

A method of solving the 0–1 knapsack problem which derives from the “shrinking boundary method” is described and compared to other methods through extensive computational experimentation.

An exact search for the solution of the surrogate dual of the 0-1 bidimensional knapsack problem

Abstract The surrogate dual of the 0–1 bidimensional knapsack problem is exactly solved by an algorithm with a modified dichotomic search. The primal (or dual) optimality is proved with a finite number of iterations. A lot of numerical experiments show the efficiency of our method: its reduced number of iterations is revealed to be independent of the size of the instances.

The 0-1 bidimensional knapsack problem: Toward an efficient high-level primitive tool

Efficient codes exist for exactly solving the 0-1 knapsack problem, which is a common primitive structure in relaxation and decomposition techniques for the solution of general models. We suggest moving to a higher primitive level by using the bidimensional knapsack, which can be used to enhance linear programming or Lagrangean type classical relaxations.With the ultimate aim of providing an exact and efficient solution to the bidimensional knapsack problem, we describe here a heuristic approach based on surrogate duality. In particular, we consider the usefulness of a specific preprocessing phase before a possible enumerative phase.Extensive numerical experiments, based on test problems from the literature as well as randomly generated instances, show that our code compares favorably with the GP procedure developed by Gavish and Pirkul for the multidimensional case.

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

We present integer linear models with a polynomial number of variables and constraints for combinatorial optimization problems in graphs: optimum elementary cycles, optimum elementary paths and optimum tree problems.

Telecommunication Network Capacity Design for Uncertain Demand

The expansion of telecommunication services has increased the number of users sharing network resources. When a given service is highly demanded, some demands may be unmet due to the limited capacity of the network links. Moreover, for such demands, telecommunication operators should pay penalty costs. To avoid rejecting demands, we can install more capacities in the existing network. In this paper we report experiments on the network capacity design for uncertain demand in telecommunication networks with integer link capacities. We use Poisson demands with bandwidths given by normal or log-normal distribution functions. The expectation function is evaluated using a predetermined set of realizations of the random parameter. We model this problem as a two-stage mixed integer program, which is solved using a stochastic subgradient procedure, the Barahonas volume approach and the Benders decomposition.

An exact algorithm for the 0–1 collapsing knapsack problem

Abstract An exact algorithm is proposed for the 0–1 collapsing knapsack problem as defined by Guignard and Posner. Bounding of the number of variables equal to 1 in an optimal solution is extensively used, as well as inner- and outer-linearization of the nonlinear right-hand side of the constraint. This allows generally a drastic reduction of the feasible domain. An implicit enumeration scheme solves the problem reduced by the preprocessing phase. Computational experiments are reported on.

Lagrangean heuristics combined with reoptimization for the 0-1 bidimensional knapsack problem

First, this paper deals with lagrangean heuristics for the 0-1 bidimensional knapsack problem. A projected subgradient algorithm is performed for solving a lagrangean dual of the problem, to improve the convergence of the classical subgradient algorithm. Secondly, a local search is introduced to improve the lower bound on the value of the biknapsack produced by lagrangean heuristics. Thirdly, a variable fixing phase is embedded in the process. Finally, the sequence of 0-1 one-dimensional knapsack instances obtained from the algorithm are solved by using reoptimization techniques in order to reduce the total computational time effort. Computational results are presented.

An Incremental Branch-And-Bound Method for the Satisfiability Problem

The satisfiability problem is to check whether a set of clauses in propositional logic is satisfiable. If it is satisfiable, the incremental satisfiability problem is then to check whether satisfiability remains given additional clauses. This paper deals with an incremental branch-and-bound method which solves exactly both problems. This method includes flexible Lagrangean relaxations, metaheuristics, and judicious jumping back. This leads to an efficient implementation which compares favorably with the classical Davis-Putnam-Loveland procedure and its incremental version designed by Hooker. Numerous computational results are detailed.