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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
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