Metaheuristics for the Template Design Problem: Encoding, Symmetry and Hybridisation

TL;DR

This paper explores and analyses a wide range of metaheuristics to tackle the template design problem with the aim of assessing their suitability for finding template designs and proposes a slot-based alternative problem formulation, which represents another option other than the classical variation-based formulation of the problem.

Abstract

The template design problem (TDP) is a hard combinatorial problem with a high number of symmetries which makes solving it more complicated. A number of techniques have been proposed in the literature to optimise its resolution, ranging from complete methods to stochastic ones. However, although metaheuristics are considered efficient methods that can find enough-quality solutions at a reasonable computational cost, these techniques have not proven to be truly efficient enough to deal with this problem. This paper explores and analyses a wide range of metaheuristics to tackle the problem with the aim of assessing their suitability for finding template designs. We tackle the problem using a wide set of metaheuristics whose implementation is guided by a number of issues such as problem formulation, solution encoding, the symmetrical nature of the problem, and distinct forms of hybridisation. For the TDP, we also propose a slot-based alternative problem formulation (distinct to other slot-based proposals), which represents another option other than the classical variation-based formulation of the problem. An empirical analysis, assessing the performance of all the metaheuristics (i.e., basic, integrative and collaborative algorithms working on different search spaces and with/without symmetry breaking) shows that some of our proposals can be considered the state-of-the-art when they are applied to specific problem instances.

Figures (4)

• Figure 1: Alternative model: Example of candidate encoding for the problem instance $\text{TDP}\langle 7,2,9\rangle$.
• Figure 2: Classical model: Example of a candidate encoding (a) and a symmetrical individual (b) for some problem instance $\text{TDP}\langle 7,2,9\rangle$
• Figure 3: Alternative model: Example of an individual that is initially symmetrical to that shown in Figure \ref{['fig:TDPModB']}. Imposing the constraint shown in Equation (\ref{['dual:symmetry breaking constraint']}) removes the possibility of symmetry.
• Figure 4: Rank distribution for the integrative and cooperative techniques chosen because of their performance in the experimentation