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Modelling and Solving the Stable Marriage problem using Constraint Programming David Manlove and Gregg OMalley University Of Glasgow Department of Computing Science

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29/07/2005 2 SM Formalisation Set n men S M = {m 1, m 2, …., m n } Set n men S M = {m 1, m 2, …., m n } Set n women S W = {w 1, w 2, …., w n } Set n women S W = {w 1, w 2, …., w n } Each man ranks the women in S W in strict order of preference. Each man ranks the women in S W in strict order of preference. Each woman ranks the men in S M in strict order of preference. Each woman ranks the men in S M in strict order of preference. A matching M is a bijection between the men and women. A matching M is a bijection between the men and women. We say a (man, woman) pair (m,w) blocks M if: We say a (man, woman) pair (m,w) blocks M if: m prefers w to his partner in M, and m prefers w to his partner in M, and w prefers m to her partner in M. w prefers m to her partner in M. A matching that admits no blocking pair is said to be stable A matching that admits no blocking pair is said to be stable Can t improve by making an arrangement outside the matching. Can t improve by making an arrangement outside the matching. SM was first formalised by David Gale and Lloyd Shapley in 1962. SM was first formalised by David Gale and Lloyd Shapley in 1962.

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29/07/2005 3 Example Stable Marriage Instance 1: 2 4 1 31: 2 1 4 3 2: 3 1 4 22: 4 3 1 2 3: 2 3 1 43: 1 4 3 2 4: 4 1 3 24: 2 1 4 3 Mens preferencesWomens preferences

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29/07/2005 4 Example Stable Matching M = {(1, 4), (2, 3), (3, 2), (4, 1)} M = {(1, 4), (2, 3), (3, 2), (4, 1)} 1: 2 4 1 31: 2 1 4 3 2: 3 1 4 22: 4 3 1 2 3: 2 3 1 43: 1 4 3 2 4: 4 1 3 24: 2 1 4 3 Mens preferencesWomens preferences

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29/07/2005 5 Extended Gale-Shapley Algorithm 1. assign each person to be free; 2. while (some man m is free) 3. w := first women on m's list; 4. if (w is currently assigned to some man p) 5. assign p to be free; 6. end if 7. assign m to w; 8. foreach (m' successors w (m) ) 9. delete (m', w); 10. end loop 11. end loop Algorithm complexityO(n 2 )– so linear in size of problem instance. Algorithm complexity O(n 2 ) – so linear in size of problem instance.

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29/07/2005 6 SM and CSPs In the last few years SM and CSPs have been the focus of much attention in the literature. In the last few years SM and CSPs have been the focus of much attention in the literature. Gent et al: CP 01 with two SM encodings Gent et al: CP 01 with two SM encodings Lustig and Puget, 2001 Lustig and Puget, 2001 Green and Cohen, CP 03 Green and Cohen, CP 03 Aldershof and Carducci, 1999 Aldershof and Carducci, 1999

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29/07/2005 7 Motivation CP 01 paper by Gent et al. presented two ways to encode an instance of SM as a Constraint Satisfaction Problem (CSP). CP 01 paper by Gent et al. presented two ways to encode an instance of SM as a Constraint Satisfaction Problem (CSP). First encoding: time taken to establish AC is O(n 4 ) - the variables domains after AC corresponded to the GS-lists. First encoding: time taken to establish AC is O(n 4 ) - the variables domains after AC corresponded to the GS-lists. Second encoding: time taken to establish AC is O(n 2 ) - the variables domains after AC corresponded to a weaker structure. Second encoding: time taken to establish AC is O(n 2 ) - the variables domains after AC corresponded to a weaker structure. Can we find an encoding that finds GS-lists in O(n 2 ) time? Can we find an encoding that finds GS-lists in O(n 2 ) time? Shows that CP algorithms give rise to the same structure and the same time complexity as conventional methods. Shows that CP algorithms give rise to the same structure and the same time complexity as conventional methods. Many stable matching problems are NP-hard – can CP help us here? Many stable matching problems are NP-hard – can CP help us here?

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29/07/2005 8 Structure of SM The Gale-Shapley (GS) algorithm has two possible orientations : The Gale-Shapley (GS) algorithm has two possible orientations : man-oriented GS (MEGS) algorithm : where the men propose to the women. man-oriented GS (MEGS) algorithm : where the men propose to the women. women-oriented GS (WEGS) algorithm : where the women propose to the men. women-oriented GS (WEGS) algorithm : where the women propose to the men. Optimality properties of each algorithm: Optimality properties of each algorithm: MEGS algorithm - man-optimal stable matching M 0 - simultaneously the best possible stable matching for all men. MEGS algorithm - man-optimal stable matching M 0 - simultaneously the best possible stable matching for all men. WEGS algorithm - woman-optimal stable matching M z - simultaneously the best possible stable matching for all women. WEGS algorithm - woman-optimal stable matching M z - simultaneously the best possible stable matching for all women. Deletions that occur during an execution of either algorithm result in a set of reduced preference lists on termination of each algorithm: Deletions that occur during an execution of either algorithm result in a set of reduced preference lists on termination of each algorithm: The MGS-lists for the MEGS algorithm. The MGS-lists for the MEGS algorithm. The WGS-lists for the WEGS algorithm. The WGS-lists for the WEGS algorithm. The intersection of the MGS-lists and the WGS-lists is called the GS-lists. The intersection of the MGS-lists and the WGS-lists is called the GS-lists. The GS-lists have many important structural properties. The GS-lists have many important structural properties.

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29/07/2005 9 New CSP Encodings We present two new CSP encodings for SM. We present two new CSP encodings for SM. First encoding (n-valued) is simple and easy to understand, presents a natural way to represent SM as CSP. First encoding (n-valued) is simple and easy to understand, presents a natural way to represent SM as CSP. Second encoding (4-valued) is more complex but addresses the problem of establishing AC and finding the GS-lists in O(n 2 ). This will only briefly be mentioned here. Second encoding (4-valued) is more complex but addresses the problem of establishing AC and finding the GS-lists in O(n 2 ). This will only briefly be mentioned here.

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29/07/2005 10 n-valued Encoding Encode SM instance I with n men and n women as a CSP instance J with 2n variables. Encode SM instance I with n men and n women as a CSP instance J with 2n variables. For each man m i S M we introduce a variable x i in J. For each man m i S M we introduce a variable x i in J. For each woman w j S W we introduce a variable y j in J. For each woman w j S W we introduce a variable y j in J. The initial domain for each variable is: The initial domain for each variable is: dom( x i ) = dom( y j ) = {1, 2,…, n}. dom( x i ) = dom( y j ) = {1, 2,…, n}. Constraints: Constraints: 1. x i p y j q ( 1 i n, 1 p n ) 2. y j q x i p ( 1 j n, 1 q n ) 3. y j q x i p ( 1 j n, 1 q n ) 4. x i p y j q ( 1 i n, 1 p n ) In Constraints 1 and 4, j is such that rank( m i, w j ) = p and also rank( w j, m i ) = q. In Constraints 2 and 3; i is such that rank( w j, m i ) = q and also rank( m i, w j ) = p.

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29/07/2005 11 n-valued Properties and Structure We prove that after AC propagation the variables domains correspond to the GS-lists. We prove that after AC propagation the variables domains correspond to the GS-lists. AC propagation in encoding takes O(n 3 ) time. AC propagation in encoding takes O(n 3 ) time. We also prove that we can enumerate all stable matchings in a failure free manner. We also prove that we can enumerate all stable matchings in a failure free manner.

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29/07/2005 12 4-valued Encoding Far more complex, extends second encoding in CP 01 paper. Far more complex, extends second encoding in CP 01 paper. Name arises from the fact that each variables domain contains 4 values Name arises from the fact that each variables domain contains 4 values 1 - is always in the domain, ensures domain never becomes empty. 1 - is always in the domain, ensures domain never becomes empty. 0 - corresponds to a proposal. 0 - corresponds to a proposal. 2 - corresponds to a deletion during the MEGS algorithm. 2 - corresponds to a deletion during the MEGS algorithm. 3 - corresponds to a deletion during the WEGS algorithm. 3 - corresponds to a deletion during the WEGS algorithm. Gives us the GS-lists after AC propagation. Gives us the GS-lists after AC propagation. AC can be established in O(n 2 ) time. AC can be established in O(n 2 ) time. Failure free enumeration of all stable matchings also holds. Failure free enumeration of all stable matchings also holds.

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29/07/2005 13 Summary Presented two new encodings for SM: Presented two new encodings for SM: Time complexities O(n 3 ) and O(n 2 ). Time complexities O(n 3 ) and O(n 2 ). AC propagation finds the GS-lists. AC propagation finds the GS-lists. Can be extended to SM with incomplete lists. Can be extended to SM with incomplete lists. First encoding has been extended to Hospitals / Residents problem (HR) First encoding has been extended to Hospitals / Residents problem (HR) Will be presented at workshop in CP 05. Will be presented at workshop in CP 05. Extension to NP-hard variants Extension to NP-hard variants HR with couples HR with couples HR with ties under weak stability HR with ties under weak stability

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