An algorithm for determining the K-best solutions of the one-dimensional Knapsack problem

Abstract

In this work we present an enumerative scheme for determining the K-best solutions (K > 1) of the one dimensional knapsack problem. If n is the total number of different items and b is the knapsack's capacity, the computational complexity of the proposed scheme is bounded by O(Knb) with memory requirements bounded by O(nb). The algorithm was implemented in a workstation and computational tests for varying values of the parameters were performed.<br>Neste trabalho apresenta-se um esquema enumerativo para se determinar as K-melhores (K > 1) soluções para o problema da mochila unidimensional. Se n é o número total de itens diferentes e b é a capacidade da mochila, a complexidade computacional do esquema proposto é limitado por O(Knb). O algoritmo foi implementado em uma estação de trabalho e testes computacionais foram realizados variando-se diferentes parâmetros do problema

Checkerboard pattern: proposals for its generation

“…The generation of more than one solution to the knapsack problem is hardly treated in the learned literature. The works of Lawler (1972) and Wolsey (1973) consider the generation of the K-best solutions to discrete optimization problems in general and, more recently, the works by Soma (1987, 1990) and Yanasse, Soma, and Maculan (2000) focused on knapsack problems. The fact that K is unknown limits further the use of the available methods.…”

“…To generate the k-best solutions to (KP1) we implemented Yanasse, Soma and Maculan's (2000) algorithm, which we will refer as YSM algorithm. Without much effort this algorithm allows the inclusion of additional constraints to the basic knapsack problem.…”

The Kth Traveling Salesman Problem is Pseudopolynomial when TSP is polynomial

“…KthTSP is motivated by searching near optimal solutions with some special properties: when in addition of the TSP comstraints, "there are some other wich might be difficult to consider explicitly in a mathematical model, or if considered, would increase largely the size of the model. By finding the best, second best, ..., Kth best solution, we are able to sequentially verify these solutions with respect to the additional constraints and stop when a solution that satisfies all of them is found" [19]. Another motivation is that if, for any reason, the route of the best solution is unavailable, then alternate solutions (routes) are desirable [16].…”

Implications of k-best modular product design solutions to global manufacturing

“…Brucker and Hamacher [8] have developed k-optimal solution sets for polynomially solvable scheduling problems. Yanasse et al [9] have presented an algorithm for k-best solutions of the one-dimensional knapsack problem. Van der Poort et al [10] have solved the k-best travelling salesman problem.…”