Matching couples with Scarf’s algorithm

Abstract

Scarf's algorithm [18] provides fractional core elements for NTU-games. Biró and Fleiner [3] showed that Scarf's algorithm can be extended for capacitated NTU-games. In this setting agents can be involved in more than one coalition at a time, cooperations may be performed with different intensities up to some limits, and the contribution of the agents can also differ in a coalition. The fractional stable solutions for the above model, produced by the extended Scarf algorithm, are called stable allocations. In this paper we apply this solution concept for the Hospitals Residents problem with Couples (HRC). This is one of the most important general stable matching problems due to its relevant applications, also well-known to be NP-hard. We show that if a stable allocation yielded by the Scarf algorithm turns out to be integral then it provides a stable matching for an instance of HRC, so this method can be used as a heuristic. In an experimental study, we compare this method with other heuristics constructed for HRC that are applied in practice in the American and Scottish resident allocation programs, respectively. Our main finding is that the Scarf algorithm outperforms all the other known heuristics when the proportion of couples is high.

Stable Matching in Large Economies

“…In these models, a finite set of workers contracts with a finite set of firms for time shares or for probabilities with which they match. Probabilistic matching is often used in allocation problems without money, such as school choice, while time share models have been proposed as a solution to labor matching markets in which part‐time jobs are available (see Biró, Fleiner, and Irving (), for instance).…”

Fractional solutions for capacitated NTU-games, with applications to stable matchings

“…The problem of finding a core element for an NTU-game can be NP-hard for some family of games or stable matching problems, but we may well need to solve such problems in practical applications. In section 5, we illustrated how the Scarf algorithm can be used as a heuristic to solve the hospitals residents problem (detailed descriptions on our experiments are given in our follow-up paper [8]). This suggests that perhaps the same technique can be used in other applications as well.…”

“…In our follow-up paper [8] we describe how the Scarf algorithm can be used efficiently in this scenario. We implemented and tested this method and compared its performance Algorithm 12 25 50 75 100 125 150 175 200 225 250 R-P (1999) 952 897 701 547 395 277 170 83 41 9 3 B-I-S (2011) 976 958 911 870 811 752 682 546 281 71 10 Scarf by B-F-I (2012) 895 813 649 532 426 356 316 261 202 174 158 …”

“…We show that an extended Scarf algorithm can find a stable allocation for this case as well. We illustrate the usage of Scarf algorithm as a heuristic for the hospitals residents problem (the complete description of this experiment can be found in our follow-up paper [8]). Finally, in Section 6, we present some important open problems and new research directions.…”

“…Suppose that we have eight residents, comprising three couples (r 1 , r 5 ), (r 2 , r 4 ) and (r 6 , r 8 ) together with two single applicants r 3 and r 7 . There are eight hospitals, h 1 , .…”