# The Power of the Weighted Sum Scalarization for Approximating Multiobjective Optimization Problems

## TL;DR

This paper investigates the effectiveness of weighted sum scalarization in approximating multiobjective optimization problems, introduces a new multi-factor approximation concept, and presents algorithms with near-optimal guarantees for minimization problems.

## Contribution

It demonstrates the power of weighted sum scalarization for approximations, introduces a novel multi-factor approximation notion, and provides algorithms achieving near-best results for multiobjective minimization.

## Key findings

- Supported solutions approximate multiobjective minimization well.
- New multi-factor approximation captures trade-offs between objectives.
- Maximization problems face inherent limitations in supported solutions.

## Abstract

We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions - both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.