Multiple Knapsack Research Articles

A new dominance procedure for combinatorial optimization problems

It is known that the effectiveness of the branch and bound algorithms for combinatorial optimization problems can be improved through dominance criteria which allow fathomings of large solution subsets. We describe a new dominance procedure which overcomes some of the drawbacks of the commonly used dominance criteria. An application to the Multiple Knapsack Problem and some computational results are also reported.

Algorithm 632: A program for the 0–1 multiple knapsack problem

Algorithm 632: A program for the 0–1 multiple knapsack problem

Cryptanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature scheme

The basic Merkle-Hellman additive trapdoor knapsack public-key cryptosystem was recently shown to be insecure, and attacks have also been developed on stronger variants of it, such as the Graham-Shamir system and the iterated knapsack cryptosystem. It is shown that some simple variants of another Merkle-Hellman system, the multiplicative knapsack cryptosystem, are insecure. It is also shown that the Shamir fast signature scheme can be broken quickly. Similar attacks can also be used to break the Schöbi-Massey authentication scheme. These attacks have not been rigorously proved to succeed, but heuristic arguments and empirical evidence indicate that they work on systems of practical size.

A Bound and Bound algorithm for the zero-one multiple knapsack problem

By the term “Bound and Bound” we define a particular tree-search technique for the ILP, which, for a maximization problem, makes use of a lower bound to determine the branches to follow in the decision tree. This technique is applied to the solution of the Zero-One Multiple Knapsack Problem and an algorithm is derived; an illustrative example of the procedure is provided. We present extensive computational results showing that the method is capable of solving problems up to 4 knapsacks and 200 variables with running times considerably smaller than those of the most commonly utilized algorithms.

Heuristic algorithms for the multiple knapsack problem

Vorhanden ist eine Menge von Gegenstanden mit gegebenem Gewicht und gegebenem Wert, ferner eine Menge von Rucksacken mit gegebener Tragfahigkeit. Die Gegenstande sollen so in die Rucksacke gepackt werden, das die Tragfahigkeit der einzelnen Rucksacke nicht uberschritten wird und der Gesamtwert der verwendeten Gegenstande moglichst gros ist. Die besten bekannten Algorithmen liefern mit vertretbarem Rechenaufwand, Losungen fur hochstens 200 Gegenstande und 4 Rucksacke. Aber praktische, Anwendungen wie z. B. Teilungs-, Lagerhaltungs- und Ladeaufgaben benotigen oft eine grosere Zahl von Variablen und verlangen, daher die Verwendung von Heuristik. Der vorliegende Artikel erlautert Methoden, mit deren Hilfe sich suboptimale Losungen des Mehr-Rucksack-Problems finden lassen. Numerische Experimente mit Problemen verschiedener, Grose zeigen, das sich die verschiedenen Algorithmen zufriedenstellend verhalten, sowohl hinsichtlich der Laufzeiten als auch bezuglich der Gute der gefundenen Losungen. Eine RORTRAN-IV-Version der Algorithmen ist beigefugt.

Solution of the zero-one multiple knapsack problem

The zero-one multiple knapsack problem is an extension of the zero-one single knapsack problem to the case where several knapsacks have to be filled and each element can be assigned to only one knapsack. This paper presents a branch and bound scheme for the solution of the problem; algorithms are then obtained by utilizing different branching strategies and bounding procedures. The computational performance of the proposed algorithm is analyzed and comparisons are made with the methods recently proposed by Hung and Fisk [4].