A Constraint Programming Approach to the Hospitals / Residents Problem By 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 R1 R2 R3 : R2 R3 R1 : R2 R1 R3 : H1 H2 H1 H2 We have n residentsand m hospitals Each resident ranks the m hospitals And each hospital ranks the n residents Objective : To find a matching of residents to hospitals Such 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 R1 R2 R3 : R2 R3 R1 : R2 R1 R3 : H1 H2 H1 H2 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 R1 R2 R3 : R2 R3 R1 : R2 R1 R3 : H1 H2 H1 H2 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 R1 R2 R3 : R2 R3 R1 : R2 R1 R3 : H1 H2 H1 H2 If a hospital has capacity > 1 (2) (1) H1a H1b H2 (1) : R2 R3 R1 : R2 R1 R3 : H1a H1b H2 It can be cloned into c hospitals with capacity 1 We then expand the residents preference lists This is now a stable marriage instance Which 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 50/13/4100/20/51k/100/105k/250/2020k/550/3750k/1.2k/42 Cloned CBM HR HRN

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

Any Questions?