Specialised Binary Constraint for the Stable Marriage Problem with Ties and incomplete preference listsBy Chris Unsworth and Patrick Prosser

Contents

The Problem Representing Ties Specialised Binary Constraint Computational Comparison Conclusion Questions

The Stable Marriage Problem

Men Women

BobIanJon

: Ian Jon Bob: Jon Ian Bob : Bob Jon Ian

: Sue Jan Liz: Liz Jan Sue: Jan Sue Liz

JanLizSue

We have n menand n women

Each man ranks the n womenAnd each woman ranks the n men

Objective :To find a matching of men to womenSuch that the matching is Stable

The Stable Marriage Problem

Men Women

BobIanJon

: Ian Jon Bob: Jon Ian Bob : Bob Jon Ian

: Sue Jan Liz: Liz Jan Sue: Jan Sue Liz

JanLizSue

A MatchingBut not a stable one

Bob and Sue would rather be matched to eachother than to there assigned partners

In this matching Bob and Sue are a Blocking pairA matching is only stable iff it contains no Blocking pairs

Ties and incomplete Preference lists

Men

Women

AlfBobTomIanJim

: Tom Alf Bob Ian: Ian (Alf Bob Jim) : (Alf Ian) Tom Bob: Tom (Jim Ian Bob) Alf: Ian Jim (Tom Bob)

: Zoe (Ann Liz) Joe: Liz Jes (Ann Zoe): (Ann Jes Liz Zoe): Ann Jes Liz Zoe Joe: Joe Zoe Jes

AnnJoeLizZoeJes

Ties indicate indifference (shown in brackets)

Incomplete list indicate some potential partners are unacceptable

Representing Ties

Each entry represented as a Triple Absolute potion in the list (ties broken

arbitrarily) Absolute potion of the first person in the

tie Absolute potion of the last person in the

tie

Zoe : Tom (Jim Ian Bob) Alf {1,1,1} {2,2,4} {2,3,4} {2,4,4}

{5,5,5}

Problem properties

Different levels of stability Here we use weak stability (details

in the paper) All instances have contain a

weakly stable matching can be found in polynomial time

Different size matching To find the largest is NP-Hard

Previous Constraint Encoding

2n Integer variables (z1..zn, zn+1..z2n) Initial domains of 1 .. L

(L = length of preference list) Domain values represent preferences

i.e. Zi = 3 : person i matched to 3rd choice

O(n2) constraints one for each zi,zj pair

where 1 ≤ i ≤ n < j ≤ 2n Explicit list of allowed tuples Each O(n2) in size

Propagates in O(n4) time Takes O(n4) space

New constraint Encoding Same 2n variables O(n2) specialised constraints

one for each zi,zj pair where 1 ≤ i ≤ n < j ≤ 2n

Two methods init()

Called when initialised (details in paper) remVal(z,a)

Called when a is removed from z (details in paper)

Propagates in O(n3) time Takes O(n2) space

Conclusion & Future work New specialised binary constraint

for SMTI Significant performance increase

over previous constraint solutions

n-ary constraint Empirical comparison

Questions?

Thank you