Strong Stability in the Hospitals/Residents Problem Rob Irving, David Manlove and Sandy Scott University of Glasgow Department of Computing Science Supported by EPSRC grant GR/R84597/01, Nuffield Foundation award NUF-NAL-02, and RSE / SEETLLD Personal Research Fellowship

Hospitals/Residents problem (HR): Motivation Graduating medical students or residents seek hospital appointments Free-for-all markets are chaotic 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

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 r finds h acceptable if h appears on r’s preference list Each hospital h has ph posts, and ranks in strict order those residents who find h acceptable A matching M is a subset of R×H such that: (r,h)M implies that r finds h acceptable Each resident r is assigned at most one hospital Each hospital h is assigned at most ph residents

An instance of HR r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 r1 r3 r5 h2:2: r3 r2 r1 r4 r5 h3:1: r4 r5 r1 r3 r6

A matching in HR r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 r1 r3 r5 h2:2: r3 r2 r1 r4 r5 h3:1: r4 r5 r1 r3 r6

Stable matchings in HR Matching M is stable if M admits no blocking pair (r,h) is a blocking pair of matching M if: r finds h acceptable and either r is unmatched in M or r prefers h to his assigned hospital in M and either h is undersubscribed in M or h prefers r to its worst resident assigned in M

A blocking pair r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 r1 r3 r5 h2:2: r3 r2 r1 r4 r5 h3:1: r4 r5 r1 r3 r6 (r4, h2) is a blocking pair of matching M since: r4 finds h2 acceptable r4 is unmatched in M h2 prefers r4 to its worst resident assigned in M

A stable matching r1: h2 h3 h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 r1 r3 r5 h2:2: r3 r2 r1 r4 r5 h3:1: r4 r5 r1 r3 r6 Example shows that, in a stable matching, one or more residents may be unmatched one or more hospitals may be undersubscribed

Algorithmic results for HR Every instance of HR admits at least one stable matching Such a matching can be found in linear time using the Gale / Shapley algorithm Resident-oriented version finds the resident-optimal stable matching Each resident obtains the best hospital that he could obtain in any stable matching Hospital-oriented version finds the hospital-optimal stable matching Each hospital obtains the best set of residents that it could obtain in any stable matching

The Rural Hospitals Theorem For a given instance of HR, there may be more than one stable matching, but: All stable matchings have the same size The same set of residents are matched in all stable matchings Any hospital that is undersubscribed in one stable matching has exactly same residents in all stable matchings This is the Rural Hospitals Theorem

Ties in the preference lists Participants may wish to express ties in their preferences lists: h1 : r7 (r1 r3) r5 version of HR with ties is HRT more general form of indifference involves partial orders version of HR with partial orders is HRP Three stability definitions are possible: Weak stability Strong stability Super-stability

An instance of HRT r1:(h2 h3) h1 h1:3: r2 (r1 r3) r5 r2: h2 h1

Weak stability Matching M is weakly stable if M admits no blocking pair (r,h) is a blocking pair of matching M if: r finds h acceptable either r is unmatched in M or r strictly prefers h to his assigned hospital in M either h is undersubscribed in M or h strictly prefers r to its worst resident assigned in M That is, if (r,h) joined together, both would be better off

A weakly stable matching in HRT r1:(h2 h3) h1 r2: h2 h1 r3: h3 (h1 h2) r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1:(r4 r5) (r1 r3 r6) Weakly stable matching has size 5

A second weakly stable matching in HRT r1:(h2 h3) h1 r2: h2 h1 r3: h3 (h1 h2) r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1:(r4 r5) (r1 r3 r6) Weakly stable matching has size 6

Weak stability in HRT: algorithmic results Weakly stable matching always exists and can be found in linear time Gusfield and Irving, 1989 Weakly stable matchings can have different sizes Problem of finding a maximum cardinality weakly stable matching is NP-hard, even if each hospital has 1 post (i.e. instance of stable marriage problem with ties and incomplete lists) Iwama, Manlove, Miyazaki and Morita, ICALP ’99 Result holds even if the ties occur on one side only, at most one tie per list, and each tie is of length 2 Manlove, Irving, Iwama, Miyazaki, Morita, TCS 2002 Result also holds for minimum weakly stable matchings

Weak stability: lower bounds Problem of finding a maximum cardinality weakly stable matching in HRT is APX-complete, even if each hospital has 1 post Halldórsson, Iwama, Miyazaki, Morita, LATIN 2002 Ties on both sides, ties of length 2 Ties on one side only, ties of length ≤3 Halldórsson, Iwama, Miyazaki, Yanagisawa, ESA 2003 Ties on one side only, ties of length 2 NP-hard to approximate maximum within 21/19 Halldórsson, Irving, Iwama, Manlove, Miyazaki, Morita, Scott, TCS 2003 Ties on one side only, preference lists of constant length Result also holds for minimum weakly stable matchings

Weak stability: upper bounds Problem of finding a maximum or minimum cardinality weakly stable matching is approximable within 2 Manlove, Irving, Iwama, Miyazaki, Morita, TCS 2002 If each hospital has 1 post, problem of finding a maximum cardinality weakly stable matching is approximable within: 2/(1+L-2) – ties on one side only, ties of length ≤L 13/7 – ties on both sides, ties of length 2 Halldórsson, Iwama, Miyazaki, Yanagisawa, ESA 2003 10/7 (expected) – ties on one side only, ≤1 tie per list, ties of length 2 7/4 (expected) – ties on both sides, ≤n ties, ties of length 2 Halldórsson, Iwama, Miyazaki, Yanagisawa, COCOON 2003 Let s+ and s- denote the sizes of a maximum and minimum cardinality weakly stable matching, and let t be the number of lists with ties. If each hospital has 1 post, any weakly stable matching M satisfies: s+ - t ≤ |M| ≤ s- + t Halldórsson, Irving, Iwama, Manlove, Miyazaki, Morita, Scott, TCS 2003

Super-stability Matching M is super-stable if M admits no blocking pair (r,h) is a blocking pair of matching M if: r finds h acceptable either r is unmatched in M or r strictly prefers h to his assigned hospital in M or r is indifferent between them either h is undersubscribed in M or h strictly prefers r to its worst resident assigned in M or h is indifferent between them That is, if (r,h) joined together, neither would be worse off

Super-stable matchings in HRT A matching is super-stable in an instance of HRT if it is stable in every instance of HR obtained by breaking the ties A super-stable matching is weakly stable “Rural Hospitals Theorem” also holds for HRT under super-stability Irving, Manlove, Scott, SWAT 2000 A super-stable matching may not exist: r1:(h1 h2) h1:1:(r1 r2) r2:(h1 h2) h2:1:(r1 r2)

Finding a super-stable matching Stable marriage problem with ties (i.e. HRT with |R|=|H|=n and each hospital has 1 post): O(n2) algorithm Irving, Discrete Applied Mathematics, 1994 General HRT case O(L) algorithm, where L is total length of preference lists Algorithm can be extended to HRP case Irving, Manlove, Scott, SWAT 2000

Strong stability in HRT A matching M is strongly stable unless there is an acceptable pair (r,h)M such that, if they joined together, one would be better off and the other no worse off Such a pair constitutes a blocking pair A super-stable matching is strongly stable, and a strongly stable matching is weakly stable Ties on one side only  super-stability = strong stability An instance of HRT may admit no strongly stable matching: r1: h1 h2 h1:1:(r1 r2) r2: h1 h2 h2:1: r1 r2

A blocking pair r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6 r4 and h2 form a blocking pair

Another blocking pair r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6 r1 and h3 form a blocking pair

Finding a strongly stable matching Stable marriage problem with ties (i.e. HRT with |R|=|H|=n and each hospital has 1 post): O(n4) algorithm Irving, Discrete Applied Mathematics, 1994 General HRT case O(L2) algorithm, where L is total length of preference lists HRP case Deciding whether a strongly stable matching exists is NP-complete Irving, Manlove, Scott, STACS 2003

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

An instance of HRT r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6

A provisional assignment and a dominated resident r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6

A deletion   r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6  

Another provisional assignment r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6  

Several provisional assignments r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6  

Provisional assignment graph with one bound resident

Removing a bound resident h1:(3) r2 h2:(1) r3 r4 h3:(1) r5

Reduced provisional assignment graph

Critical set r1 h2:(1) r3 r4 h3:(1) r5

Critical set   r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6  

Deletions from the critical set, end of loop iteration r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3   h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6          

Second loop iteration, starting with a provisional assignment r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3   h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6          

Several provisional assignments r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3   h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6          

Final provisional assignment graph with 4 bound residents

Removing a bound resident h1:(2) r2 h2:(2) r3 r4 h3:(1) r5

Removing a second bound resident h1:(2) r2 h2:(1) r3 r4 h3:(1) r5

Removing a third bound resident

Removing a bound resident with an additional provisional assignment

Final reduced provisional assignment graph

A feasible matching This consists of: the bound (resident,hospital) pairs (r1,h1), (r2,h2), (r3,h2), (r5,h1) unioned together with: perfect matching in the final reduced provisional assignment graph (r4,h3)

A strongly stable matching (r1,h1), (r2,h2), (r3,h2), (r4,h3), (r5,h1) r1: (h2 h3) h1 r2: h2 h1 r3: h3 h2 h1 r4: h2 h3 r5: h2 (h1 h3) r6: h3 h1:3: r2 (r1 r3) r5 h2:2: r3 r2 (r1 r4 r5) h3:1: (r4 r5) (r1 r3) r6

The algorithm in full (1) 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 provisional 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 = ;

The algorithm in full (2) 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;

Theoretical results Algorithm has complexity O(L2), where L is total length of preference lists Bounded below by complexity of finding a perfect matching in a bipartite graph Matching produced by the algorithm is resident-optimal “Rural Hospitals Theorem” holds for HRT under strong stability

Strong stability in HRP HRP under strong stability is NP-complete even if each hospital has 1 post and all pairs are 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

Conclusions O(nL) algorithm, where n is number of residents and L is total length of preference lists Kavitha, Mehlhorn, Michail, Paluch, STACS 2004 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