Approximating Biobjective Minimization Problems Using General Ordering Cones

TL;DR

This paper explores how solutions approximate biobjective minimization problems when using larger ordering cones, providing guarantees that depend on the cone's inner angle and extending known results.

Contribution

It generalizes and unifies existing approximation results for biobjective minimization using convex cones and shows limitations for maximization problems.

Findings

Approximation guarantees depend continuously on the cone's inner angle.

Supported solutions provide a 2-approximation, and the efficient set provides a 1-approximation.

No approximation guarantee is possible for maximization problems with larger cones.

Abstract

This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering $\alpha$-approximations for all closed convex ordering cones of a fixed inner angle $\gamma \in [\frac \pi 2, \pi]$, an approximation guarantee between $\alpha$ and $2 \alpha$ is achieved, which depends continuously on $\gamma$. The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.