1. Strong Stability in the Hospitals/Residents Problem Robert W. Irving, David F. Manlove and Sandy Scott University of Glasgow Department of Computing Science Supported by EPSRC grant GR/R84597/01 and Nuffield Foundation Award NUF-NAL-02

2. 8/4/03BCTCS 192 Hospitals/Residents problem (HR): Motivation Graduating medical students or residents seek hospital appointments Centralised matching schemes are in operation Schemes produce stable matchings of residents to hospitals – National Resident Matching Program (US) – other large-scale matching schemes, both educational and vocational

3. 8/4/03BCTCS 193 Hospitals/Residents problem (HR): Definition a set H of hospitals, a set R of residents each resident r ranks a subset of H in strict order of preference each hospital h has p h posts, and ranks in strict order those residents who have ranked it a matching M is a subset of the acceptable pairs of R H such that |{h: (r,h) M}| 1 for all r and |{r: (r,h) M}| p h for all h

4. 8/4/03BCTCS 194 An instance of HR r 1 : h 2 h 3 h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 h 1 h 3 r 6 : h 3 h 1 :3: r 2 r 1 r 3 r 5 h 2 :2: r 3 r 2 r 1 r 4 r 5 h 3 :1: r 4 r 5 r 1 r 3 r 6

5. 8/4/03BCTCS 195 A matching in HR r 1 : h 2 h 3 h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 h 1 h 3 r 6 : h 3 h 1 :3: r 2 r 1 r 3 r 5 h 2 :2: r 3 r 2 r 1 r 4 r 5 h 3 :1: r 4 r 5 r 1 r 3 r 6

6. 8/4/03BCTCS 196 Indifference in the ranking ties: h 1 : r 7 (r 1 r 3 ) r 5 version of HR with ties is HRT more general form of indifference involves partial orders version of HR with partial orders is HRP

7. 8/4/03BCTCS 197 An instance of HRT r 1 : (h 2 h 3 ) h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 (h 1 h 3 ) r 6 : h 3 h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 (r 1 r 4 r 5 ) h 3 :1: (r 4 r 5 ) (r 1 r 3 ) r 6

8. 8/4/03BCTCS 198 r 1 : (h 2 h 3 ) h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 (h 1 h 3 ) r 6 : h 3 h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 (r 1 r 4 r 5 ) h 3 :1: (r 4 r 5 ) (r 1 r 3 ) r 6 A matching in HRT

9. 8/4/03BCTCS 199 A blocking pair r 1 : (h 2 h 3 ) h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 (h 1 h 3 ) r 6 : h 3 h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 (r 1 r 4 r 5 ) h 3 :1: (r 4 r 5 ) (r 1 r 3 ) r 6 r 4 and h 2 form a blocking pair

10. 8/4/03BCTCS 1910 Stability a matching M is stable unless there is an acceptable pair (r,h) M such that, if they joined together both would be better off (weak stability) neither would be worse off (super-stability) one would be better off and the other no worse off (strong stability) such a pair constitutes a blocking pair hereafter consider only strong stability

11. 8/4/03BCTCS 1911 Another blocking pair r 1 : (h 2 h 3 ) h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 (h 1 h 3 ) r 6 : h 3 h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 (r 1 r 4 r 5 ) h 3 :1: (r 4 r 5 ) (r 1 r 3 ) r 6 r 1 and h 3 form a blocking pair

12. 8/4/03BCTCS 1912 A strongly stable matching r 1 : (h 2 h 3 ) h 1 r 2 : h 2 h 1 r 3 : h 3 h 2 h 1 r 4 : h 2 h 3 r 5 : h 2 (h 1 h 3 ) r 6 : h 3 h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 (r 1 r 4 r 5 ) h 3 :1: (r 4 r 5 ) (r 1 r 3 ) r 6

13. 8/4/03BCTCS 1913 State of the art for HRT / HRP weak stability: –weakly stable matching always exists –efficient algorithm (Gale and Shapley (AMM, 1962), Gusfield and Irving (MIT Press, 1989)) –matchings may vary in size (Manlove et al. (TCS, 2002)) –many NP-hard problems, including finding largest weakly stable matching (Iwama et al. (ICALP, 1999), Manlove et al. (TCS, 2002))

14. 8/4/03BCTCS 1914 State of the art for HRT / HRP super-stability –super-stable matching may or may not exist –efficient algorithm (Irving, Manlove and Scott (SWAT, 2000)) strong stability –strongly stable matching may or may not exist –efficient algorithm for HRT –in contrast, problem is NP-complete in HRP (Irving, Manlove and Scott (STACS, 2003))

15. 8/4/03BCTCS 1915 The algorithm in brief repeat provisionally assign all free residents to hospitals at head of list form reduced provisional assignment graph find critical set of residents and make corresponding deletions until critical set is empty form a feasible matching check if feasible matching is strongly stable

16. 8/4/03BCTCS 1916 Properties of the algorithm algorithm has complexity O(a 2 ), where a is the number of acceptable pairs bounded below by complexity of finding a perfect matching in a bipartite graph matching produced by the algorithm is resident-optimal same set of residents matched and posts filled in every strongly stable matching

17. 8/4/03BCTCS 1917 Strong stability in HRP HRP under strong stability is NP-complete –even if all hospitals have just one post, and every pair is acceptable reduction from RESTRICTED 3-SAT: –Boolean formula B in CNF where each variable v occurs in exactly two clauses as literal v, and exactly two clauses as literal ~v

18. 8/4/03BCTCS 1918 Open problems find a weakly stable matching with minimum number of strongly stable blocking pairs size of strongly stable matchings relative to possible sizes of weakly stable matchings hospital-oriented algorithm