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 andNuffield Foundation Award NUF-NAL-02

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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

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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 ph 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}| ph for all h

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An instance of HR

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

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A matching in HR

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

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Indifference in the ranking

• ties: 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

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An instance of HRT

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

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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 matching in HRT

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A blocking pair

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

r4 and h2 form a blocking pair

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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

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Another blocking pair

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

r1 and h3 form a blocking pair

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A strongly stable matching

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

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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))

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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))

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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

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Properties of the algorithm

• algorithm has complexity O(a2), 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

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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

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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