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By Chris Unsworth and Patrick Prosser

Specialised Binary Constraint for the Stable Marriage Problem with Ties and incomplete preference lists. By Chris Unsworth and Patrick Prosser. Contents. The Problem Representing Ties Specialised Binary Constraint Computational Comparison Conclusion Questions.

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By Chris Unsworth and Patrick Prosser

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  1. Specialised Binary Constraint for the Stable Marriage Problem with Ties and incomplete preference lists By Chris Unsworth and Patrick Prosser

  2. Contents • The Problem • Representing Ties • Specialised Binary Constraint • Computational Comparison • Conclusion • Questions

  3. The Stable Marriage Problem We have n men and n women Each man ranks the n women And each woman ranks the n men Men Women Jan Liz Sue : Ian Jon Bob : Jon Ian Bob : Bob Jon Ian Bob Ian Jon : Sue Jan Liz : Liz Jan Sue : Jan Sue Liz Objective : To find a matching of men to women Such that the matching is Stable

  4. The Stable Marriage Problem A Matching But not a stable one  Bob and Sue would rather be matched to each other than to there assigned partners Men Women Jan Liz Sue : Ian Jon Bob : Jon Ian Bob : Bob Jon Ian Bob Ian Jon : Sue Jan Liz : Liz Jan Sue : Jan Sue Liz In this matching Bob and Sue are a Blocking pair A matching is only stable iff it contains no Blocking pairs

  5. Ties and incomplete Preference lists Men Women Ann Joe Liz Zoe Jes : Tom Alf Bob Ian : Ian (Alf Bob Jim) : (Alf Ian) Tom Bob : Tom (Jim Ian Bob) Alf : Ian Jim (Tom Bob) Alf Bob Tom Ian Jim : Zoe (Ann Liz) Joe : Liz Jes (Ann Zoe) : (Ann Jes Liz Zoe) : Ann Jes Liz Zoe Joe : Joe Zoe Jes Ties indicate indifference (shown in brackets) Incomplete list indicate some potential partners are unacceptable

  6. 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}

  7. 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

  8. 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

  9. 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

  10. Conclusion & Future work • New specialised binary constraint for SMTI • Significant performance increase over previous constraint solutions • n-ary constraint • Empirical comparison

  11. Questions? • Thank you

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