The multiple knapsack problem with grouped items aims to maximize rewards by assigning groups of among knapsacks, without exceeding capacities. Either all in a group are assigned or none at all. We study the bi-criteria variation problem, where capacities can be exceeded and second objective is minimize maximum capacity. propose approximation algorithms that run pseudo-polynomial time guarantee...

Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete diffe...

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of th...

In this paper we study and solve two different variants of static knapsack problems with random weights: The stochastic knapsack problem with simple recourse as well the stochastic knapsack problem with probabilistic constraint. Special regard is given the corresponding continuous problems and three different problem solving methods are presented. The resolution of the continuous problems serve...

It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in O(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it can be solved in pseudo-polynomial time. We show however that the dynamic programming appro...

We consider the average-case performance of a well-known approximation algorithm for the 0/1 knapsack problem, the Decreasing Density Greedy (DDG) algorithm. Let Un = {u1, . . . , un} be a set of n items, with each item ui having a size si and a profit pi, and Kn be the capacity of the knapsack. Given an instance of the 0/1 knapsack problem, let PL denote the total profit of an optimal solution...

We construct three public key knapsack cryptosystems. Standard knapsack cryptosystems hide easy instances of the knapsack problem and have been broken. The systems considered in the article face this problem: They hide a random (possibly hard) instance of the knapsack problem. We provide both complexity results (size of the key, time needed to encypher/decypher...) and experimental results. Sec...

We formulate the NP-hard n-dimensional knapsack feasibility problem as an equivalent absolute value equation (AVE) in an n-dimensional noninteger real variable space and propose a finite succession of linear programs for solving the AVE. Exact solutions are obtained for 1880 out of 2000 randomly generated consecutive knapsack feasibility problems with dimensions between 500 and one million. For...