Reference Point Methods and Approximation in Multicriteria Optimization

TL;DR

This paper explores the approximation of reference point solutions in multicriteria optimization, establishing their polynomial equivalence to Pareto set approximation and extending single-criterion algorithms to the multicriteria context.

Contribution

It demonstrates the polynomial equivalence between approximating reference point solutions and Pareto sets, and adapts existing single-criterion algorithms to multicriteria optimization.

Findings

Approximation of reference point solutions is polynomially equivalent to Pareto set approximation.

Single-criterion algorithms like dynamic programming and LP-rounding can be extended to multicriteria problems.

Applicable to problems like Set Cover and machine scheduling.

Abstract

Operations research applications often pose multicriteria problems. Mathematical research on multicriteria problems predominantly revolves around the set of Pareto optimal solutions, while in practice, methods that output a single solution are more widespread. In real-world multicriteria optimization, reference point methods are widely used and successful examples of such methods. A reference point solution is the solution closest to a given reference point in the objective space. We study the approximation of reference point solutions. In particular, we establish that approximating reference point solutions is polynomially equivalent to approximating the Pareto set. Complementing these results, we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization. In particular, we lift the link between dynamic programming and FPTAS, as well as oblivious LP-rounding techniques. The latter applies, e.g., to Set Cover and several machine scheduling problems.