A Framework for Handling and Exploiting Symmetry in Benders' Decomposition

TL;DR

This paper develops a systematic approach to detect and exploit symmetry in Benders' decomposition, improving efficiency by reducing oracle calls through graph-based symmetry detection and tailored cut handling techniques.

Contribution

It introduces a novel theory for symmetry detection within Benders' decomposition using graph models, enabling classical symmetry handling methods to be applied.

Findings

Symmetry handling accelerates Benders' decomposition in bin packing and scheduling.

Graph-based symmetry detection effectively identifies symmetries in the BD framework.

Cut aggregation reduces the number of oracle executions significantly.

Abstract

Benders' decomposition (BD) is a framework for solving optimization problems by removing some variables and modeling their contribution to the original problem via so-called Benders cuts. While many advanced optimization techniques can be applied in a BD framework, one central technique has not been applied systematically in BD: symmetry handling. The main reason for this is that Benders cuts are not known explicitly but only generated via a separation oracle. In this work, we close this gap by developing a theory of symmetry detection within the BD framework. To this end, we introduce a tailored family of graphs that capture the symmetry information of both the Benders master problem and the Benders oracles. Once symmetries of these graphs are known, which can be found by established techniques, classical symmetry handling approaches become available to accelerate BD. We complement these approaches by devising techniques for the separation and aggregation of symmetric Benders cuts by means of tailored separation routines and extended formulations. Both substantially reduce the number of executions of the separation oracles. In a numerical study, we show the effect of both symmetry handling and cut aggregation for bin packing and scheduling problems.