1. Pelvic floor Specialised Constraints for Stable Matching Problems Contact details: 17 Lilybank Gardens Glasgow G12 8QQ chrisu@dcs.gla.ac.uk The Problem The Stable Matching problems we have been looking at are the Stable Marriage problem (SMP) and a generalisation of this problem known as the Hospital/Residents problem (HRP). In SMP we have a set of n men and a set of n women. Each man ranks all the women into a strictly ordered preference list, and the women do the same with the men. Specialised Constraint Solutions Algorithmic Solutions Constraint Solutions Men Women Bob Ian Jon : Ian Jon Bob : Jon Ian Bob : Bob Jon Ian : Sue Jan Liz : Liz Jan Sue : Jan Sue Liz Jan Liz Sue The objective is then to find a matching of men to woman such that the matching is stable. By matching we mean a bijection of men to women, and by stable we mean that there is not a man and woman that are not matched to each other in the matching that would rather be matched to each other than their assigned partners. Men Women Bob Ian Jon : Ian Jon Bob : Jon Ian Bob : Bob Jon Ian : Sue Jan Liz : Liz Jan Sue : Jan Sue Liz Jan Liz Sue In HRP instead of Men we have residents wishing to be assigned to hospitals, instead of women we have hospitals wishing to have residents assigned to them. Each hospital has a capacity, meaning that one or more residents can be assigned to a single hospital as long as the hospitals capacity is not exceeded. Optimal algorithms have been proposed for both of these problems. The algorithms reduce the preference lists to a fixed point such that a solution can easily be found, that solution is a highly biased. The SMP algorithm returns either the man-optimal or woman-optimal solution. In the man-optimal solution all men are assigned their best possible partner from all stable matchings and all women will be assigned there worst possible match. Several constraint solutions have been proposed for SMP. From a simplest of which is a model that has an explicit constraint for each potential couple. It has been proven that enforcing arc-consistency on this model reduces the variable domains to a state which is equivalent to the fixed point reached by the SMP algorithm. It has also been proven that because of this all stable matchings can be found in a failure free search. This model can easily be adapted to solve many harder variants were stability is only part of a richer problem. Arc- consistency can be enforced on this model in O(n 4 ) time. A more complex constraint model has also been proposed which uses boolean variables and standard toolbox constraints. When made arc-consistent the variables in this model represent the bounds of the fixed point reached by the SMP algorithm. This model can also be used to find all stable matchings in a failure free search. This model can also be adapted to solver impure SMP’s. The space and time complexity of enforcing arc-consistency on this model is O(n 2 ) which is optimal to the size of the problem. Although this model has the same optimal space and time complexities as the SMP algorithm in practice there is a large gap between the performance of the algorithm and this constraint model. By Chris Unsworth We have proposed four specialised constraint solutions for these stable matching problems, two for SMP and two for HRP. This idea behind these specialised constraints is to try and reduce the performance gap between the constraint solutions and the relevant algorithms whilst retaining the flexibility of the constraint solutions. The first specialised constraint for SMP is a specialised binary constraint (SM2) which works in much the same way as the explicit constraint model. Instead of the lookup table with each constraint in the explicit constraint, this specialised binary constraint has a propagation method which gets called each time a variable looses a value. Arc-consistency can be enforced on a model using this constraint in O(n 3 ) time. The second SMP constraint is a specialise n-ary constraint (SMN). This constraint works in the same way as SM2 except that the propagation method has access to all variables rather than only two, therefore only one such constraint is needed to model a SMP instance. Arc-consistency can be enforced on a model using this constraint in O(n 2 ) time which is optimal to the problem size. The two HRP constraint work in much the same way in that there is a specialised binary constraint (HR2) that acts between a single hospital and a single resident, and an specialised n-ary constraint that constrains all hospitals and all residents. Size n model451002004006008001000 Explicit8.95------ Bool0.251.24.4---- SM20.160.230.51.824.218.0212.47 SMN0.010.020.060.210.510.952.11 In this table we give the average time taken to enforce arc-consistency on 100 randomly generated SMP instances using the various constraint models. An entry – in the table denotes that a memory error occurred. As can be seen both the specialised constraints for SMP (SM2 and SMN) significantly improve on the traditional constraint models (Explicit and Bool). A similar performance improvement was also observed with the HRP constraints.