Short Rational Generating Functions For Multiobjective Linear Integer Programming

TL;DR

This paper introduces algorithms leveraging Barvinok's rational functions to efficiently generate all nondominated solutions in multiobjective linear integer programming, with proven polynomial complexity in fixed dimensions.

Contribution

It presents a novel algorithmic approach using rational functions to encode and enumerate nondominated solutions, with theoretical complexity analysis and practical implementation.

Findings

Encoding nondominated solutions is polynomial in fixed dimensions.

The algorithm provides polynomial delay enumeration.

Implementation demonstrates practical usefulness for multiobjective ILPs.

Abstract

This paper presents algorithms for solving multiobjective integer programming problems. The algorithm uses Barvinok's rational functions of the polytope that defines the feasible region and provides as output the entire set of nondominated solutions for the problem. Theoretical complexity results on the algorithm are provided in the paper. Specifically, we prove that encoding the entire set of nondominated solutions of the problem is polynomially doable, when the dimension of the decision space is fixed. In addition, we provide polynomial delay algorithms for enumerating this set. An implementation of the algorithm shows that it is useful for solving multiobjective integer linear programs.