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 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 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 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 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 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 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 A matching in 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

9 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 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 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 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 State of the art 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 State of the art 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 –here we present an efficient algorithm for HRT –in contrast, show problem is NP-complete in HRP –(Irving, Manlove and Scott (STACS, 2003))

15 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 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

17 A provisional assignment and a dominated resident 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

18 A deletion 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 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 )

19 Another provisional assignment 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 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 )

20 Several provisional assignments 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 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 )

21 The provisional assignment graph with one bound resident r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(3) h 2 :(2) h 3 :(1)

22 Removing a bound resident r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(3) h 2 :(1) h 3 :(1)

23 The reduced provisional assignment graph r3r3 r4r4 r5r5 r1r1 h 2 :(1) h 3 :(1)

24 The critical set 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 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 )

25 Deletions from the critical set, end of loop iteration r 1 : h 1 r 2 : h 2 h 1 r 3 : h 2 h 1 r 4 : h 3 r 5 : (h 1 h 3 ) r 6 : h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 h 3 :1: (r 4 r 5 )

26 Second loop iteration, starting with a provisional assignment r 1 : h 1 r 2 : h 2 h 1 r 3 : h 2 h 1 r 4 : h 3 r 5 : (h 1 h 3 ) r 6 : h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 h 3 :1: (r 4 r 5 )

27 Several provisional assignments r 1 : h 1 r 2 : h 2 h 1 r 3 : h 2 h 1 r 4 : h 3 r 5 : (h 1 h 3 ) r 6 : h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 h 3 :1: (r 4 r 5 )

28 The final provisional assignment graph with four bound residents r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(3) h 2 :(2) h 3 :(1)

29 Removing a bound resident r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(2) h 2 :(2) h 3 :(1)

30 Removing another bound resident r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(2) h 2 :(1) h 3 :(1)

31 Removing a third bound resident r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(2) h 2 :(0) h 3 :(1)

32 Removing a bound resident with an additional provisional assignment r2r2 r3r3 r4r4 r5r5 r1r1 h 1 :(1) h 2 :(0) h 3 :(1)

33 The reduced final provisional assignment graph r4r4 h 3 :(1)

34 A cancelled assignment r 1 : h 1 r 2 : h 2 h 1 r 3 : h 2 h 1 r 4 : h 3 r 5 : (h 1 h 3 ) r 6 : h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 h 3 :1: (r 4 r 5 )

35 A feasible matching r 1 : h 1 r 2 : h 2 h 1 r 3 : h 2 h 1 r 4 : h 3 r 5 : (h 1 h 3 ) r 6 : h 1 :3: r 2 (r 1 r 3 ) r 5 h 2 :2: r 3 r 2 h 3 :1: (r 4 r 5 )

36 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

37 repeat { while some resident r is free and has a non-empty list for each hospital h at the head of r’s list { provisionally assign r to h; if h is fully-subscribed or over-subscribed { for each resident r' dominated on h’s list delete the pair (r',h); } } form the reduced assignment graph; find the critical set Z of residents; for each hospital h  N(Z) for each resident r in the tail of h’s list delete the pair (r,h); }until Z =  ;

38 let G be the final provisional assignment graph; let M be a feasible matching in G; if M is strongly stable output M; else no strongly stable matching exists;

39 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

40 Strong Stability in HRP HRP 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 variable v, and exactly two clauses as ~v

41 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