Accommodating tied preferences in the SFAS Matching Scheme

ii Acknowledgements I would like to thank my project supervisor, Dr Rob Irving, whose endless patience, advice and assistance has been greatly appreciated. I would also like to thank Dr Helen Purchase for her efforts in securing a faculty scholarship that has enabled me to attend the MSc Information Technology course over the last twelve months. Abstract The Scottish Foundation Allocation Scheme (SFAS) is a centralised matching scheme which is responsible for matching graduating students from the four Scottish university medical schools to their first hospital foundation training programmes. Students and hospitals submit 'preference lists' which rank their preferred foundation programmes or students respectively before a sophisticated algorithm attempts to allocate each student to a hospital foundation programme, based on these preference lists [18]. This is an example of the Hospitals-Residents problem with one-sided ties. Irving and Manlove's paper, " Finding Large Stable Matchings " [11] outlined a recent empirical study undertaken to compare different heuristics solving instances of the Hospitals-Residents problem when ties are expressed on one side. The study suggested that algorithms employing " smart " tie-breaking strategies consistently find larger stable matchings than purely arbitrary tie breaking algorithms. As a logical progression, it was of interest to compare random tie-breaking and " smart " tie-breaking strategies when ties are permitted on both hospital and resident preference lists. Furthermore, in the context of SFAS, we were interested to investigate whether encouraging students to introduce ties in their preference lists would yield larger matchings. An application was developed which will facilitate a full-scale empirical evaluation comparing a random tie-breaking algorithm with the Király [14] algorithm which employs a " smart " tie-breaking strategy to solve instances of the Hospitals-Residents problem with two-sided ties. The application is believed to be the first known practical implementation of the Király algorithm for the Hospitals-Residents problem with unrestricted ties. From the results of a small-scale empirical study carried out as part of the project, initial results suggest that the Király algorithm produces consistently larger matchings, on average, than those produced by the algorithm breaking ties in a random way. Furthermore, results suggest that introducing ties on both sides' preference lists results in larger maximum-sized matchings, on average, than when ties are only permitted in hospital preference lists.