Strong%20Stability%20in%20the%20Hospitals/Residents%20Problem

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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
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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

• h13 r2 r1 r3 r5
• h22 r3 r2 r1 r4 r5
• h31 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

• h13 r2 r1 r3 r5
• h22 r3 r2 r1 r4 r5
• h31 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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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A matching in HRT

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6 h3

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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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))

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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))

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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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An instance of HRT

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6 h3

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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A deletion

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3)

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Another provisional assignment

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3)

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Several provisional assignments

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3)

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The provisional assignment graph with one bound
resident
r1
h1(3)
r2
h2(2)
r3
r4
h3(1)
r5
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Removing a bound resident
r1
h1(3)
r2
h2(1)
r3
r4
h3(1)
r5
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The reduced provisional assignment graph
r1
h2(1)
r3
r4
h3(1)
r5
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The critical set

• r1 (h2 h3) h1
• r2 h2 h1
• r3 h3 h2 h1
• r4 h2 h3
• r5 h2 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3)

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Deletions from the critical set, end of loop
iteration

• r1 h1
• r2 h2 h1
• r3 h2 h1
• r4 h3
• r5 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2
• h31 (r4 r5)

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Second loop iteration, starting with a
provisional assignment

• r1 h1
• r2 h2 h1
• r3 h2 h1
• r4 h3
• r5 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2
• h31 (r4 r5)

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Several provisional assignments

• r1 h1
• r2 h2 h1
• r3 h2 h1
• r4 h3
• r5 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2
• h31 (r4 r5)

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The final provisional assignment graph with four
bound residents
r1
h1(3)
r2
h2(2)
r3
r4
h3(1)
r5
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Removing a bound resident
r1
h1(2)
r2
h2(2)
r3
r4
h3(1)
r5
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Removing another bound resident
r1
h1(2)
r2
h2(1)
r3
r4
h3(1)
r5
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Removing a third bound resident
r1
h1(2)
r2
h2(0)
r3
r4
h3(1)
r5
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Removing a bound resident with an additional
provisional assignment
r1
h1(1)
r2
h2(0)
r3
r4
h3(1)
r5
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The reduced final provisional assignment graph
r4
h3(1)
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A cancelled assignment

• r1 h1
• r2 h2 h1
• r3 h2 h1
• r4 h3
• r5 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2
• h31 (r4 r5)

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A feasible matching

• r1 h1
• r2 h2 h1
• r3 h2 h1
• r4 h3
• r5 (h1 h3)
• r6

• h13 r2 (r1 r3) r5
• h22 r3 r2
• h31 (r4 r5)

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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

• h13 r2 (r1 r3) r5
• h22 r3 r2 (r1 r4 r5)
• h31 (r4 r5) (r1 r3) r6

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repeat while some resident r is free and has a
non-empty list for each hospital h at the head
of rs list provisionally assign r to
h if h is fully-subscribed or over-subscribed
for each resident r' dominated on hs
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 hs list delete
the pair (r,h) until Z ?
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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
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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 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

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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