Ian Miguel

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.

Global Constraints for Lexicographic Orderings

We propose some global constraints for lexicographic orderings on vectors of variables. These constraints are very useful for breaking a certain kind of symmetry arising in matrices of decision variables. We show that decomposing such constraints carries a penalty either in the amount or the cost of constraint propagation. We therefore present a global consistency algorithm which enforces a lexicographic ordering between two vectors of n variables in O(nb) time, where b is the cost of adjusting the bounds of a variable. The algorithm can be modified very slightly to enforce a strict lexicographic ordering. Our experimental results on a number of domains (balanced incomplete block design, social golfer, and sports tournament scheduling) confirm the efficiency and value of these new global constraints.

Essence: A constraint language for specifying combinatorial problems

Essence is a formal language for specifying combinatorial problems in a manner similar to natural rigorous specifications that use a mixture of natural language and discrete mathematics. Essence provides a high level of abstraction, much of which is the consequence of the provision of decision variables whose values can be combinatorial objects, such as tuples, sets, multisets, relations, partitions and functions. Essence also allows these combinatorial objects to be nested to arbitrary depth, providing for example sets of partitions, sets of sets of partitions, and so forth. Therefore, a problem that requires finding a complex combinatorial object can be specified directly by using a decision variable whose type is precisely that combinatorial object.

An evaluation of machine learning in algorithm selection for search problems

Machine learning is an established method of selecting algorithms to solve hard search problems. Despite this, to date no systematic comparison and evaluation of the different techniques has been performed and the performance of existing systems has not been critically compared with other approaches. We compare the performance of a large number of different machine learning techniques from different machine learning methodologies on five data sets of hard algorithm selection problems from the literature. In addition to well-established approaches, for the first time we also apply statistical relational learning to this problem. We demonstrate that there is significant scope for improvement both compared with existing systems and in general. To guide practitioners, we close by giving clear recommendations as to which machine learning techniques are likely to achieve good performance in the context of algorithm selection problems. In particular, we show that linear regression and alternating decision trees have a very high probability of achieving better performance than always selecting the single best algorithm.

Propagation algorithms for lexicographic ordering constraints

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finite-domain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq.

CGRASS: a system for transforming constraint satisfaction problems

Experts at modelling constraint satisfaction problems (CSPs) carefully choose model transformations to reduce greatly the amount of effort that is required to solve a problem by systematic search. It is a considerable challenge to automate such transformations and to identify which transformations are useful. Transformations include adding constraints that are implied by other constraints, adding constraints that eliminate symmetrical solutions, removing redundant constraints and replacing constraints with their logical equivalents. This paper describes the CGRASS (Constraint Generation And Symmetry-breaking) system that can improve a problem model by automatically performing transformations of these kinds. We focus here on transforming individual CSP instances. Experiments on the Golomb ruler problem suggest that producing good problem formulations solely by transforming problem instances is, generally, infeasible. We argue that, in certain cases, it is better to transform the problem class than individual instances and, furthermore, it can sometimes be better to transform formulations of a problem that are more abstract than a CSP.

Conditional symmetry breaking

We introduce the study of Conditional symmetry breaking in constraint programming. This arises in a sub-problem of a constraint satisfaction problem, where the sub-problem satisfies some condition under which additional symetries hold. Conditional symmetry can cause redundancy in a systematic search for solutions. Breaking this symmetry is an important part of solving a constraint satisfaction problem effectively. We demonstrate experimentally that three methods, well-known for breaking unconditional symmetries, can be applied to conditional symmetries. These are: adding conditional symmetry-breaking constraints, reformulating the problem to remove the symmetry, and augmenting the search process to break the conditional symmetry dynamically through the use of a variant of Symmetry Breaking by Dominance Detection (SBDD). We thank Alan Frisch and Chris Jefferson. Ian Gent is supported by a Royal Society of Edinburgh SEELLD/RSE Support Research Fellowship. Ian Miguel is supported by a UK Royal Academy of Engineering/EPSRC Research Fellowship. This material is based in part on works supported by the Science Foundation Ireland under Grant No. 00/PI.1/C075.

Constraint Generation via Automated Theory Formation

Adding constraints to a basic CSP model can significantly reduce search,e.g. for Golomb rulers [6]. The generation process is usually performed by hand, although some recent work has focused on automatically generating symmetry breaking constraints [4]and (less so)on generating implied constraints [5]. We describe an approach to generating implied,symmetry breaking and specialisation constraints and apply this technique to quasigroup construction [10].

Constraints for breaking more row and column symmetries

Constraint programs containing a matrix of two (or more) dimensions of decision variables often have row and column symmetries: in any assignment to the variables the rows can be swapped and the columns can be swapped without affecting whether or not the assignment is a solution. This introduces an enormous amount of redundancy when searching a space of partial assignments. It has been shown previously that one can remove consistently some of these symmetries by extending such a program with constraints that require the rows and columns to be lexicographically ordered. This paper identifies and studies the properties of a new additional constraint—the first row is less than or equal to all permutations of all other rows—that can be added consistently to break even more symmetries. Two alternative implementations of this stronger symmetry-breaking method are investigated, one of which employs a new algorithm that in time linear in the size of the matrix enforces the constraint that the first row is less than or equal to all permutations of all other rows. It is demonstrated experimentally that our method for breaking more symmetries substantially reduces search effort.

Fuzzy rrDFCSP and planning

Constraint satisfaction is a fundamental Artificial Intelligence technique for knowledge representation and inference. However, the formulation of a static constraint satisfaction problem (CSP) with hard, imperative constraints is insufficient to model many real problems. Fuzzy constraint satisfaction provides a more graded viewpoint. Priorities and preferences are placed on individual constraints and aggregated via fuzzy conjunction to obtain a satisfaction degree for a solution to the problem. This paper examines methods for solving an important instance of dynamic flexible constraint satisfaction (DFCSP) combining fuzzy CSP and restriction/relaxation based dynamic CSP: fuzzy rrDFCSP. This allows the modelling of complex situations where both the set of constraints may change over time and there is flexibility inherent in the definition of the problem. This paper also presents a means by which classical planning can be extended via fuzzy sets to enable flexible goals and preferences to be placed on the use of planning operators. A range of plans can be produced, trading compromises made versus the length of the plan. The flexible planning operators are close in definition to fuzzy constraints. Hence, through a hierarchical decomposition of the planning graph, the work shows how flexible planning reduces to the solution of a set of fuzzy rrDFCSPs.