Constrained Stochastic Submodular Maximization with State-Dependent Costs
This work addresses optimization challenges in resource allocation under uncertainty, such as in recommendation systems or sensor placement, but is incremental as it builds on existing submodular maximization frameworks.
The paper tackles the problem of constrained stochastic submodular maximization with state-dependent costs, where item states are unknown until selected and must satisfy both state-dependent and state-independent constraints, and presents a constant approximate solution.
In this paper, we study the constrained stochastic submodular maximization problem with state-dependent costs. The input of our problem is a set of items whose states (i.e., the marginal contribution and the cost of an item) are drawn from a known probability distribution. The only way to know the realized state of an item is to select that item. We consider two constraints, i.e., \emph{inner} and \emph{outer} constraints. Recall that each item has a state-dependent cost, and the inner constraint states that the total \emph{realized} cost of all selected items must not exceed a give budget. Thus, inner constraint is state-dependent. The outer constraint, one the other hand, is state-independent. It can be represented as a downward-closed family of sets of selected items regardless of their states. Our objective is to maximize the objective function subject to both inner and outer constraints. Under the assumption that larger cost indicates larger "utility", we present a constant approximate solution to this problem.