On Search Space Symmetry in Partitioning Problems

Abstract

Many problems consist in splitting a set of objects so that each part verifies some properties. In practice, a partitioning is often encoded by an array mapping each object to its group numbering. In fact, the group number of a object does not really matter, and one can simply rename each part to obtain a new encoding. That is what we call the symmetry of the search space in a partitioning problem. This property may be prejudicial for methods such as evolutionary algorithms (EA) which require some diversity during their executions.

This article aims at providing a theoretical framework for breaking this symmetry. We define an equivalence relation on the encoding space. This leads us to define a non-trivial search space which eliminates symmetry. We define polynomially computable tools such as equality test, a neighborhood operator and a metric applied on the set of partitioning.