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 PTASs for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTASs. We finally illustrate better PTASs and FPTASs for the subset sum case of the problem in which profits and weights coincide.