Integer Programming Subject to Monomial Constraints

TL;DR: It is shown that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time.

Abstract: We investigate integer programs containing constraints of the type $\prod_{i\in I}x_i^{\alpha_i}=b$. Due to the number-theoretic nature of these constraints, standard methods based on linear algebra cannot be applied directly. Instead, we present a reformulation resulting in integer programs with linear constraints and polynomial objective functions, using prime decompositions of the right-hand sides $b$. Moreover, we show that minimizing a linear objective function with nonnegative coefficients over bivariate constraints is possible in polynomial time.