Linear Decision Tree Policies for Integer Linear Programs

TL;DR

This paper introduces linear decision tree policies for integer linear programs, enabling fast retrieval of optimal solutions for varying cost vectors after an initial synthesis, with theoretical guarantees and practical benefits.

Contribution

It presents a polynomial-time synthesis method for linear decision tree policies that efficiently solve ILPs for multiple cost vectors, improving solution speed.

Findings

Policies perform optimal solutions in polynomial time.

Synthesis is computationally intensive but enables faster repeated queries.

Approach offers a new offline-online paradigm for ILPs.

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.