a constraint programming approach to the hospitals / residents problem by david manlove, gregg...
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![Page 1: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/1.jpg)
A Constraint Programming Approach to theHospitals / Residents ProblemBy David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth
![Page 2: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/2.jpg)
Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 4: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/4.jpg)
The Hospital/Residents Problem This is a real world problem The National Resident Matching
Program (NRMP) in the US 31,000 residents matched to 2,300
hospitals The Canadian Resident Matching
Service (CaRMS) The Scottish PRHO Allocation
scheme (SPA)
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The Hospital/Residents Problem
Residents Hospitals
R1R2R3
: R2 R3 R1: R2 R1 R3
: H1 H2: H1 H2: H1 H2
H1H2
We have n residentsand m hospitals
Each resident ranks the m hospitalsAnd each hospital ranks the n residents
Objective :To find a matching of residents to hospitalsSuch that the matching is Stable
(2)(1)
Each hospital has a capacity c
And the hospital capacities not exceeded
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The Hospital/Residents Problem
Residents Hospitals
R1R2R3
: R2 R3 R1: R2 R1 R3
: H1 H2: H1 H2: H1 H2
H1H2
A matching
R3 and H1 would both be better off if they were matched to each other
(2)(1)
But not a stable one
A matching is only stable iff it contains no Blocking pairs
In this matching R3 and H1 are a Blocking pair
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The Hospital/Residents Problem
Residents Hospitals
R1R2R3
: R2 R3 R1: R2 R1 R3
: H1 H2: H1 H2: H1 H2
H1H2
A stable matching
(2)(1)
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 9: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/9.jpg)
The Algorithms
Two Algorithms Resident-Oriented (RGS) Hospital-Oriented (HGS)
Both reach a fixed point RGS-lists HGS-lists Union of these is GS-lists
Both run in O(L) time and require O(nm) space
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 11: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/11.jpg)
Cloned Solution
Residents Hospitals
R1R2R3
: R2 R3 R1: R2 R1 R3
: H1 H2: H1 H2: H1 H2
H1H2
If a hospital has capacity > 1
(2)(1)
H1aH1bH2
(1)(1)(1)
: R2 R3 R1: R2 R3 R1: R2 R1 R3
: H1a H1b H2: H1a H1b H2: H1a H1b H2
It can be cloned into c hospitals with capacity 1We then expand the residents preference lists
This is now a stable marriage instanceWhich can be solved by any stable marriage solution
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 13: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/13.jpg)
Constraint Based Model (CBM) a variable for each of the n
Residents each with a domain (1 .. m)
C variables for each of the m Hospitals each with a domain (1 .. n)
O(Lc) standard “toolbox” constraints Takes O(Lc(n+m)) time to enforce
AC Takes O(Lc) space
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 15: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/15.jpg)
Specialised Binary Constraint (HR2) a variable for each of the n
Residents each with a domain (1 .. m)
a variables for each of the m Hospitals each with a domain (1 .. n)
O(L) Specialised Binary constraints Details are in the paper
Takes O(Lc(n+m)) time to enforce AC
Takes O(nm) space
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 17: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/17.jpg)
Specialised N-ary Constraint (HRN) a variable for each of the n
Residents each with a domain (1 .. m)
a variables for each of the m Hospitals each with a domain (1 .. n)
1 Specialised n-ary constraints Details are in the paper
Takes O(Lc) time to enforce AC Takes O(nm) space
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 19: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/19.jpg)
Computational Comparison
50/13/4 100/20/5 1k/100/10 5k/250/20 20k/550/37
50k/1.2k/42
Cloned 5.84 - - - - -
CBM 0.24 0.36 4.75 - - -
HR2 0.15 0.18 0.88 9.91 112 -
HRN 0.12 0.15 0.22 0.53 1.42 4.2
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 21: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/21.jpg)
Conclusion
We have proposed three new constraint solutions to the Hospital/Residents problem A model that uses toolbox
constraints A specialised binary constraint A specialised n-ary constraint
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Contents The Hospital/Residents Problem The Algorithms Cloned Solution Constraint Based Model (CBM) Specialised Binary Constraint (HR2) Specialised N-ary Constraint (HRN) Computational Comparison Conclusion Questions
![Page 23: A Constraint Programming Approach to the Hospitals / Residents Problem By David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth](https://reader030.vdocuments.mx/reader030/viewer/2022032702/56649f455503460f94c668b0/html5/thumbnails/23.jpg)
Questions
Any Questions?