Weak Symmetries in Problem Formulations

Abstract

Unlike symmetries weak symmetries act only on a subset of the variables and/or respect only a subset of the constraints of the problem. Therefore, weak symmetries preserve the state of feasibility only with respect to the subset of variables they act on and only for the constraints they respect. This means if two solutions are symmetric under the weak symmetry they yield different full solutions with potentially different feasibility states.

But weak symmetries cannot be simply broken, since this would result in a loss of solutions that cannot be derived afterwards. Therefore we propose a modelling technique that uses additional variables (SymVars) and constraints that enable us to express symmetric states of a solution. The idea is to decompose a problem P in a way such that the variables and constraints respected by the weak symmetry is present in one sub-problem P ₁ and the rest in the sub-problem P ₂. This way the weak symmetry acts as a common symmetry on P ₁. The additional variables and constraints form a new sub-problem P _sym that is incorporated and the solving order is to find a solution to P ₁, consider a symmetric equivalent by P _sym and pass the solution to P ₂ which finds a solution for the whole problem. By doing so the symmetry on P ₁ can be broken.

Although additional variables are introduced which extends the search space symmetry breaking enables us to reduce the search effort. Whether symmetry breaking does compensate the extension of the search space by the additional variables depends on the problem and the search heuristic. But as soon as a solution for P ₁ is found the whole equivalence class of solutions can be considered via P _sym .

Weak symmetries occur in various problems.They can be found in real-life problems (especially optimisation problems where the weak symmetry does not respect the objective value) as well as in in classic problem formulations like the magic square problem [1] or extensions of problems like the diagonal latin square [2] or the weighted magic square problem [3].