Brahim Hnich

Breaking Row and Column Symmetries in Matrix Models

We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.

Specification, implementation, and deployment of components

Clarifying common terminology and exploring component-based relationships.

A Meta-heuristic for Subset Problems

In constraint solvers, variable and value ordering heuristics are used to finetune the performance of the underlying search and propagation algorithms. However, few guidelines have been proposed for when to choose what heuristic among the wealth of existing ones. Empirical studies have established that this would be very hard, as none of these heuristics outperforms all the other ones on all instances of all problems (for an otherwise fixed solver). The best heuristic varies not only between problems, but even between different instances of the same problem. Taking heed of the popular dictum If you cant beat them, join them! we devise a practical meta-heuristic that automatically chooses, at run-time, the best available heuristic for the instance at hand. It is applicable to an entire class of NP-complete subset problems.

Towards Inferring Labelling Heuristics for CSP Application Domains

Many real-life problems can be represented as constraint satisfaction problems (CSPs) and then be solved using constraint solvers, in which labelling heuristics are used to fine-tune the performance of the underlying search algorithm. However, few guidelines have been proposed for the application domains of these heuristics. If a mapping between application domains and heuristics is known to the solver, then modellers can -- if they wish so -- be relieved from figuring out which heuristic to indicate or implement. Instead of inferring the application domains of (known) heuristics, we advocate inferring (known or new) heuristics for application domains. Our approach is to first formalise a CSP application domain as a family of models, so as to exhibit the generic constraint store for all models in that family. Second, family-specific labelling heuristics are inferred by analysing the interaction of a given search algorithm with this generic constraint store. We illustrate our approach on a domain of subset problems.

Models of Injection Problems

There are many problems that can be modelled as injection problems. These problems can be scheduling problems, combinatorial graph problems, cryptarithmetic puzzles, etc. Injection problems can be modelled as constraint satisfaction problems. The straightforward formulation would be to have as many variables as the elements of the source set that range over the target set, which captures a total function. To enforce that the function is injective, we would need to state an alldifferent constraint among all the variables. Dual variables can also be used along with the primal ones and linked through channeling constraints. We propose three different ways of achieveing this, as well as we add some implied constraints. The proposed models of injection problems are compared using the constraint tightness parameterized by the level of local consistency being enforced [2]. We proved that, with respect to arc-consistency a single primal alldifferent constraint is tighter than channeling constraints together with the implied or the dual not-equals constraints, but that the channeling constraints alone are as tight as the primal not-equals constraints. Both these gaps can lead to an exponential reduction in search cost when MAC or MGAC are used. The theoretical results showed that occurs constraints on dual variables are redundant, so we can safely discard them. The asymptotic analysis added details to the theoretical results. We conclude that it is safe to discard some of the models because they achieve less pruning than other models at the same cost. However, we keep a model employing primal and dual variables even though it achieves the same amount of pruning as a primal model at a higher cost because it might allow the development of cheap value ordering heuristics. Experimental results on a sport scheduling problem confirmed that MGAC on channeling and implied constraints outperformed MAC on primal not-equals constraints, and could be competitive with maintaining GAC on a primal alldifferent constraint.