On the complexity and approximation of the maximum expected value all-or-nothing subset

An unconstrained nonlinear binary optimization problem of selecting a maximum expected value subset of items is considered. Each item is associated with a profit and probability. Each of the items succeeds or fails independently with the given probabilities, and the profit is obtained in the event that all selected items succeed. The objective is to select a subset that maximizes the total value times the product of probabilities of the chosen items. The problem is proven NP-hard by a nontrivial reduction from subset sum. Then, after proving bounds on the probabilities of items selected in an optimal solution, we develop a fully polynomial time approximation scheme (FPTAS) for this problem. Next we consider an extension to maximizing the same objective over a more general downward closed set system. Here the problem is reduced to the solution of a sequence of constrained linear problems. This method is then used to propose a polynomial time approximation scheme (PTAS) for maximum expected value all-or-nothing matching in a graph and all-or-nothing matroid intersection using results for the corresponding constrained linear problem variants.