Linear Decision Tree Policies for Integer Linear Programs

Abstract

We study optimal decision policies for integer linear programs with a fixed feasible set and varying cost vectors, represented as linear decision trees. Once synthesized for a given feasible set, they return an optimal solution for any queried cost vector through a sequence of linear tests. We show that there exists a policy performing this operation in a polynomial number of arithmetic operations in the worst case. Along with this theoretical guarantee, we develop a practical construction framework to synthesize policies within a specific subclass of linear decision trees. Our computational experiments show that, although policy synthesis can be time-intensive, it allows retrieving optimal solutions orders of magnitude faster than classical and specialized solution methods on repeated queries. Overall, this paradigm provides a new perspective on the complexity of integer linear programs and offers an offline--online approach for solving them.