Generalized budgeted submodular set function maximization

In the generalized budgeted submodular set function maximization problem, we are given a ground set of elements and a set of bins. Each bin has its own cost and the cost of each element depends on its associated bin. The goal is to find a subset of elements along with an associated set of bins such that the overall costs of both is at most a given budget, and the profit is maximized. We present an algorithm that guarantees a  -approximation, where is the approximation factor of an algorithm for a sub-problem. If the costs satisfy a specific condition, we provide a polynomial-time algorithm that gives us , while for the general case we design an algorithm with.

We extend our results providing a bi-criterion approximation algorithm where we can spend an extra budget up to a factor
to guarantee a -approximation.