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 N-ary Constraint

(HRN) Versatility 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)

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

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

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

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

Versatility

Resident-exchange-stable NP-Complete Constraint for each r1,r2,h1,h2

combination Such that:

r1 prefers h1 to h2

r2 prefers h2 to h1

r1=h2 r2≠h1

r1=h2 r2≠h1

Versatility

Forbidden pairs Linear time solvable decision

problem Not all instances are solvable Relaxed optimisation problem

No poly-time algorithm Add {0,1} variable for each forbidden

pair Minimise the sum

Versatility

Groups No poly-time algorithm Constraint for each group

r1=hi r2=hi

Ties in preference lists

Conclusion

We have proposed three new constraint solutions to the Hospital/Residents problem A Reformulation technique A model that uses toolbox

constraints A specialised n-ary constraint

Demonstration of versatility

Questions

Any Questions?