a constraint programming approach to the hospitals / residents problem by david manlove, gregg...
TRANSCRIPT
A Constraint Programming Approach to theHospitals / Residents ProblemBy David Manlove, Gregg O’Malley, Patrick Prosser and Chris Unsworth
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
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
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)
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Questions
Any Questions?