Automatic Generation of Polynomial Symmetry Breaking Constraints
This work addresses the problem of computational inefficiency due to symmetry in integer programming for optimization practitioners, but it is incremental as it builds on existing symmetry breaking methods with a new algebraic approach.
The authors tackled the problem of symmetry in integer programming, which causes redundant search, by proposing an algebraic method to automatically generate polynomial symmetry breaking constraints; they tested the approach on near half-capacity 0-1 bin packing instances and found that simple symmetry breakers, especially those combining few variables and permutations, most consistently reduced work time.
Symmetry in integer programming causes redundant search and is often handled with symmetry breaking constraints that remove as many equivalent solutions as possible. We propose an algebraic method which allows to generate a random family of polynomial inequalities which can be used as symmetry breakers. The method requires as input an arbitrary base polynomial and a group of permutations which is specific to the integer program. The computations can be easily carried out in any major symbolic computation software. In order to test our approach, we describe a case study on near half-capacity 0-1 bin packing instances which exhibit substantial symmetries. We statically generate random quadratic breakers and add them to a baseline integer programming problem which we then solve with Gurobi. It turns out that simple symmetry breakers, especially combining few variables and permutations, most consistently reduce work time.