# On the Complexity and Approximation of the Maximum Expected Value All-or-Nothing Subset

## TL;DR

This paper studies a complex nonlinear optimization problem involving selecting items with probabilities and profits to maximize expected value, proves its NP-hardness, and develops an efficient approximation scheme.

## Contribution

It introduces the first FPTAS for the maximum expected value subset problem, addressing its computational complexity.

## Key findings

- The problem is NP-hard via reduction from subset sum.
- An FPTAS is developed for the problem.
- The approximation scheme provides near-optimal solutions efficiently.

## Abstract

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 we develop a fully polynomial time approximation scheme (FPTAS) for this problem.