Approximability issues for unconstrained and constrained maximization of half-product related functions

We address the Boolean programming problem of maximizing a half-product function, with and without a linear knapsack constraint. Maximizing the half-product can be done in polynomial time. Adding a knapsack constraint makes the problem non-approximable within a constant factor, provided that the coefficients in the linear part of the function are negative. For maximizing a function with positive coefficients in the linear part we develop a fully polynomial-time approximation scheme.