A Polynomial-Time Inner Approximation Algorithm for Multi-Objective Optimization
Levin Nemesch, Stefan Ruzika, Clemens Thielen, Alina Wittmann
TL;DR
The authors address the intractability of obtaining all non-dominated images in multi-objective MOMILPs by targeting $(1+\varepsilon)$-convex approximation sets. They introduce a grid-agnostic inner-approximation algorithm that leverages Csirmaz's skeleton approach and a plane-separating oracle built from weighted-sum scalarization, enabling adaptive construction without grid dependence. The method achieves polynomial-time running time under a polynomial-time oracle and empirically outperforms the previous best on assignment, knapsack, and symmetric metric TSP instances. The approach extends to generalizations with per-objective tolerances and connects to parametric optimization, offering a practical tool when grid-based convex-approximation is undesirable.
Abstract
In multi-objective optimization, the problem of finding all non-dominated images is often intractable. However, for any multiplicative factor greater than one, an approximation set can be constructed in polynomial time for many problems. In this paper, we use the concept of convex approximation sets: Each non-dominated image is approximated by a convex combination of images of solutions in such a set. Recently, Helfrich et al. (2024) presented a convex approximation algorithm that works in an adaptive fashion and outperforms all previously existing algorithms. We use a different approach for constructing an even more efficient adaptive algorithm for computing convex approximation sets. Our algorithm is based on the skeleton algorithm for polyhedral inner approximation by Csirmaz (2021). If the weighted sum scalarization can be solved exactly or approximately in polynomial time, our algorithm can find a convex approximation set for an approximation factor arbitrarily close to this solution quality. We demonstrate that our new algorithm significantly outperforms the current state-of-the-art algorithm from Helfrich et al. (2024) on instances of the multi-objective variants of the assignment problem, the knapsack problem, and the symmetric metric travelling salesman problem.