Approximation schemes for non-separable non-linear boolean programming problems under nested knapsack constraints

We consider a fairly general model of “take-or-leave” decision-making. Given a number of items of a particular weight, the decision-maker either takes (accepts) an item or leaves (rejects) it. We design fully polynomial-time approximation schemes (FPTASs) for optimization of a non-separable non-linear function which depends on which items are taken and which are left. The weights of the taken items are subject to nested constraints. There is a noticeable lack of approximation results on integer programming problems with non-separable functions. Most of the known positive results address special forms of quadratic functions, and in order to obtain the corresponding approximation algorithms and schemes considerable technical difficulties have to be overcome. We demonstrate how for the problem under consideration and its modifications FPTASs can be designed by using (i) the geometric rounding techniques, and (ii) methods of K-approximation sets and functions. While the latter approach leads to a faster scheme, the running times of both algorithms compare favorably with known analogues for less general problems.