Approximation algorithms for knapsack problems with cardinality constraints

We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting-plane approaches to integer linear programs. We focus our attention on approximation algorithms for the problem, describing a linear-storage Polynomial Time Approximation Scheme (PTAS) and a dynamic-programming based Fully Polynomial Time Approximation Scheme (FPTAS). The main ideas contained in our PTAS are used to derive PTAS's for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTAS's. We finally illustrate better PTAS's and FPTAS's for the subset sum case of the problem in which profits and weights coincide. (C) 2000 Elsevier Science B.V. All rights reserved.