Pareto Optima of Multicriteria Integer Linear Programs

TL;DR

This paper establishes the computational complexity of key problems in multicriteria integer linear programming with fixed dimensions, providing efficient algorithms for counting, enumerating, and approximating Pareto optima.

Contribution

It introduces polynomial-time algorithms for counting and enumerating Pareto optima and strategies, and develops approximation schemes for distance minimization problems.

Findings

Polynomial-time algorithms for counting Pareto optima and strategies

Polynomial-delay enumeration of Pareto sets under projections

Fully polynomial-time approximation scheme for Euclidean distance minimization

Abstract

We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm for arbitrary projections of the Pareto set; 3. an algorithm to minimize the distance of a Pareto optimum from a prescribed comparison point with respect to arbitrary polyhedral norms; 4. a fully polynomial-time approximation scheme for the problem of minimizing the distance of a Pareto optimum from a prescribed comparison point with respect to the Euclidean norm.