Insight Centre for Data Analytics

Recent Progress in Two-Sided StableMatching

Mohamed Sialahttp://ucc.insight-centre.org/msiala

May 2, 2018

Academic Career

• December 2011 to May 2015:• PhD in Computer science, LAAS-CNRS, INSA Toulouse,

France• Funding: CNRS, Google, Midi-Pyrenees

• Since June 2015:• Post Doctoral Researcher, Insight, Centre for Data

Analytics, University College Cork, Ireland.• Funding: Science Foundation Ireland 90% and UCC-UTRC

10%.

Insight Centre for Data Analytics May 2, 2018 Slide 2

Research Areas & Applications

Research Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Research Areas & ApplicationsResearch Areas

• Constraint programming: [CP’12, IJCAI’13, CPAIOR’14,Constraints’14, CP’15, EAAI’15, CPAIOR’16, Constraints’16,CP’17, IJCAI’17, CPAIOR’17, CPAIOR’18]

• Boolean SAT & Clause Learning:[CPAIOR’14, CP’15, CP’17, CPAIOR’17]

• Combinatorial optimisation, algorithmic complexity,operations research:[EAAI’15, Constraints’16, ICTAI’17, COCOA’17, IJCAI’17]

Applications

• Scheduling & Sequencing Problems:[CP’12, IJCAI’13, CPAIOR’14, Constraints’14, CP’15,EAAI’15, Constraints’16]

• Matching under Preferences:[CPAIOR’16, CP’17, IJCAI’17, ICTAI’17, COCOA’17]

Insight Centre for Data Analytics May 2, 2018 Slide 3

Matching Under Preferences

• They are everywhere! (doctors to hospitals, houseallocation, kidney exchange, etc)

• Robustness?

• Modularity & Flexibility of CP to solve hard problems?

Insight Centre for Data Analytics May 2, 2018 Slide 4

Matching Under Preferences

• They are everywhere! (doctors to hospitals, houseallocation, kidney exchange, etc)

• Robustness?

• Modularity & Flexibility of CP to solve hard problems?

Insight Centre for Data Analytics May 2, 2018 Slide 4

Matching Under Preferences

• They are everywhere! (doctors to hospitals, houseallocation, kidney exchange, etc)

• Robustness?

• Modularity & Flexibility of CP to solve hard problems?

Insight Centre for Data Analytics May 2, 2018 Slide 4

Matching Under Preferences

• They are everywhere! (doctors to hospitals, houseallocation, kidney exchange, etc)

• Robustness?

• Modularity & Flexibility of CP to solve hard problems?

Insight Centre for Data Analytics May 2, 2018 Slide 4

Matching Under Preferences

• They are everywhere! (doctors to hospitals, houseallocation, kidney exchange, etc)

• Robustness?

• Modularity & Flexibility of CP to solve hard problems?

Insight Centre for Data Analytics May 2, 2018 Slide 4

Matching Under Preferences

Insight Centre for Data Analytics May 2, 2018 Slide 5

Matching Under Preferences

• Assign residents to hospitals

• Every resident has a personnel preference over hospitals

• Each hospital has a preference list over residents

Insight Centre for Data Analytics May 2, 2018 Slide 5

Matching Under Preferences

• Assign students to universities

• Every student has a personnel preference overuniversities

• Each university has a preference list over students

Insight Centre for Data Analytics May 2, 2018 Slide 5

Paper: Finding Robust Solutions to Stable Marriage

Title: Finding Robust Solutions to Stable Marriage

Authors: Begum Genc, Mohamed Siala, Barry O’Sullivan,Gilles Simonin

IJCAI-’17, August 2017, Melbourne, Australia

Insight Centre for Data Analytics May 2, 2018 Slide 6

Context

Insight Centre for Data Analytics May 2, 2018 Slide 7

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Background

• A set of men U = {m1,m2, . . . ,mn1} and a set of womanW = {w1,w2, . . . ,wn2}

• Each person has an ordinal preference list over people ofthe opposite sex

• Amatching M is a one-to-one correspondence between UandW

• A matchingM is called stable if for every (m,w) ∈ M ,• ifm prefers w ′ to w , then ∃(m′,w ′) ∈ M such that w ′

prefersm′ tom• if w prefersm′ tom, then ∃(m′,w ′) ∈ M such thatm′

prefers w ′ to w

Insight Centre for Data Analytics May 2, 2018 Slide 8

Motivation

Insight Centre for Data Analytics May 2, 2018 Slide 9

Motivation

Insight Centre for Data Analytics May 2, 2018 Slide 9

Overview of the contributions

(a,b)-supermatch([Ginsberg98supermodelsand, 07-hebrard-phd])An (a, b)-supermatch is a stable matching in which if a pairs breakup it is possible to find another stable matching by changing thepartners of those a pairs and at most b other pairs

Contributions (regarding the (1,b) case)

• Verification in polynomial time

• Three models to find the most robust solution

• Experimental study on random instances

Insight Centre for Data Analytics May 2, 2018 Slide 10

Overview of the contributions

(a,b)-supermatch([Ginsberg98supermodelsand, 07-hebrard-phd])An (a, b)-supermatch is a stable matching in which if a pairs breakup it is possible to find another stable matching by changing thepartners of those a pairs and at most b other pairs

Contributions (regarding the (1,b) case)

• Verification in polynomial time

• Three models to find the most robust solution

• Experimental study on random instances

Insight Centre for Data Analytics May 2, 2018 Slide 10

Example

m0 0 6 5 2 4 1 3 w0 2 1 6 4 5 3 0m1 6 1 4 5 0 2 3 w1 0 4 3 5 2 6 1m2 6 0 3 1 5 4 2 w2 2 5 0 4 3 1 6m3 3 2 0 1 4 6 5 w3 6 1 2 3 4 0 5m4 1 2 0 3 4 5 6 w4 4 6 0 5 3 1 2m5 6 1 0 3 5 4 2 w5 3 1 2 6 5 4 0m6 2 5 0 6 4 3 1 w6 4 6 2 1 3 0 5

• This instance has 10 stable matchings

Insight Centre for Data Analytics May 2, 2018 Slide 11

Dominance Relation of Stable Matchings

Insight Centre for Data Analytics May 2, 2018 Slide 12

Dominance Relation of Stable Matchings

Insight Centre for Data Analytics May 2, 2018 Slide 12

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

〈1, 4〉

〈5, 0〉

〈6, 5〉

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

〈1, 5〉

〈5, 4〉

〈6, 0〉

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

〈1, 5〉

〈5, 4〉

〈6, 0〉

• The sequence ρ1 = [〈1, 4〉, 〈5, 0〉, 〈6, 5〉] is called a rotation

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

〈1, 5〉

〈5, 4〉

〈6, 0〉

• The sequence ρ1 = [〈1, 4〉, 〈5, 0〉, 〈6, 5〉] is called a rotation

• 〈1, 4〉 is eliminated by ρ1

Insight Centre for Data Analytics May 2, 2018 Slide 13

Rotation

• M1 : 〈0, 2〉, 〈1, 4〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 0〉, 〈6, 5〉• M2 : 〈0, 2〉, 〈1, 5〉, 〈2, 6〉, 〈3, 3〉, 〈4, 1〉, 〈5, 4〉, 〈6, 0〉

〈1, 5〉

〈5, 4〉

〈6, 0〉

• The sequence ρ1 = [〈1, 4〉, 〈5, 0〉, 〈6, 5〉] is called a rotation

• 〈1, 4〉 is eliminated by ρ1• 〈1, 5〉 is produced by ρ1

Insight Centre for Data Analytics May 2, 2018 Slide 13

The Partial Order on Rotations

Insight Centre for Data Analytics May 2, 2018 Slide 14

Rotation Graph

Insight Centre for Data Analytics May 2, 2018 Slide 15

Closed Subset

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

Insight Centre for Data Analytics May 2, 2018 Slide 16

Closed Subset

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

Insight Centre for Data Analytics May 2, 2018 Slide 16

Verification

• M : a stable matching

• b: is an integer

• S : closed subset ofM

• 〈m,w〉: couple to break-up

• ρp : rotation that produces 〈m,w〉• ρe : rotation that eliminates 〈m,w〉• SUP : The largest closed subset⊂ S that

does not include ρp

• SDOWN : The smallest closed subset⊃ Sthat includes ρe

Insight Centre for Data Analytics May 2, 2018 Slide 17

Verification

• M : a stable matching

• b: is an integer

• S : closed subset ofM

• 〈m,w〉: couple to break-up

• ρp : rotation that produces 〈m,w〉• ρe : rotation that eliminates 〈m,w〉• SUP : The largest closed subset⊂ S that

does not include ρp

• SDOWN : The smallest closed subset⊃ Sthat includes ρe

Insight Centre for Data Analytics May 2, 2018 Slide 17

Verification

• M : a stable matching

• b: is an integer

• S : closed subset ofM

• 〈m,w〉: couple to break-up

• ρp : rotation that produces 〈m,w〉• ρe : rotation that eliminates 〈m,w〉

• SUP : The largest closed subset⊂ S thatdoes not include ρp

• SDOWN : The smallest closed subset⊃ Sthat includes ρe

Insight Centre for Data Analytics May 2, 2018 Slide 17

Verification

• M : a stable matching

• b: is an integer

• S : closed subset ofM

• 〈m,w〉: couple to break-up

• ρp : rotation that produces 〈m,w〉• ρe : rotation that eliminates 〈m,w〉• SUP : The largest closed subset⊂ S that

does not include ρp

• SDOWN : The smallest closed subset⊃ Sthat includes ρe

Insight Centre for Data Analytics May 2, 2018 Slide 17

Verification

• M : a stable matching

• b: is an integer

• S : closed subset ofM

• 〈m,w〉: couple to break-up

• ρp : rotation that produces 〈m,w〉• ρe : rotation that eliminates 〈m,w〉• SUP : The largest closed subset⊂ S that

does not include ρp

• SDOWN : The smallest closed subset⊃ Sthat includes ρe

Insight Centre for Data Analytics May 2, 2018 Slide 17

Robust Solutions

ProblemGiven a SM instance, find the most robust stable matching. Thatis, find a (1,b)-supermatch such that b is minimum

NP-Hard!On the Complexity of Robust Stable Marriage, Begum Genc,Mohamed Siala, Gilles Simonin, Barry O’Sullivan, COCOA’17,December 2017, Shanghai, China.

Insight Centre for Data Analytics May 2, 2018 Slide 18

Robust Solutions

ProblemGiven a SM instance, find the most robust stable matching. Thatis, find a (1,b)-supermatch such that b is minimum

NP-Hard!On the Complexity of Robust Stable Marriage, Begum Genc,Mohamed Siala, Gilles Simonin, Barry O’Sullivan, COCOA’17,December 2017, Shanghai, China.

Insight Centre for Data Analytics May 2, 2018 Slide 18

Local Search: Key Idea

• Random solutions based on random closed subsets

• The evaluation of a solution is based on the verificationprocedure

• The neighbourhood of a solution S is defined byadding/removing one rotation to S

Insight Centre for Data Analytics May 2, 2018 Slide 19

Genetic Algorithm

• Random population based on random closed subsets

• The evaluation of a solution is based on the verificationprocedure

• Crossover: Given S1 and S2, pick at random ρ1 ∈ S1, thenadd ρ1 and all its predecessors to S2

• Mutation: Given S and a random rotation ρ, if ρ /∈ S , thenadd ρ and all its predecessors to S . Otherwise, remove ρand all its successors to S

Insight Centre for Data Analytics May 2, 2018 Slide 20

Experimental Study

0 0.2 0.4 0.6 0.8 1

0

500

1,000

Objective ratio

CPU

Tim

e

CPGALS

Insight Centre for Data Analytics May 2, 2018 Slide 21

Experimental Study

0 0.2 0.4 0.6 0.8 1

0

500

1,000

Objective ratio

CPU

Tim

e

CPGALS

Insight Centre for Data Analytics May 2, 2018 Slide 21

Experimental Study: Large Instances

0 0.2 0.4 0.6 0.8 1

0

500

1,000

Objective ratio

CPU

Tim

e

GALS

Insight Centre for Data Analytics May 2, 2018 Slide 22

Paper: Rotation-Based Formulation for StableMatching

Title: Rotation-Based Formulation for Stable Matching

Authors:Mohamed Siala and Barry O’Sullivan,

CP’17, August 2017, Melbourne, Australia

Insight Centre for Data Analytics May 2, 2018 Slide 23

Context

• Many to many stable matching as a global constraint

• Modularity and Efficiency to tackle hard variants

Insight Centre for Data Analytics May 2, 2018 Slide 24

Context

• Many to many stable matching

as a global constraint

• Modularity and Efficiency to tackle hard variants

Insight Centre for Data Analytics May 2, 2018 Slide 24

Context

• Many to many stable matching as a global constraint

• Modularity and Efficiency to tackle hard variants

Insight Centre for Data Analytics May 2, 2018 Slide 24

Context

• Many to many stable matching as a global constraint

• Modularity and Efficiency to tackle hard variants

Insight Centre for Data Analytics May 2, 2018 Slide 24

Context

• Many to many stable matching as a global constraint

• Modularity and Efficiency to tackle hard variants

Insight Centre for Data Analytics May 2, 2018 Slide 24

Key Idea: Rotation-based Reformulation

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• Express the relationship between the two sets of variables

Insight Centre for Data Analytics May 2, 2018 Slide 25

Key Idea: Rotation-based Reformulation

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• Express the relationship between the two sets of variables

Insight Centre for Data Analytics May 2, 2018 Slide 25

Key Idea: Rotation-based Reformulation

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1

• Express the relationship between the two sets of variables

Insight Centre for Data Analytics May 2, 2018 Slide 25

Key Idea: Rotation-based Reformulation

Theorem [89-sm-book, 07-mm-matching]There is a one-to-one mapping between closed subsets andstable matchings

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• Express the relationship between the two sets of variables

Insight Centre for Data Analytics May 2, 2018 Slide 25

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Important Notions & Properties

• A pair is stablewhen it belongs to a stable matching

• Some pairs are non-stable

• Some pairs are fixed

• In O(L) time, one can compute:

• M0,Mz

• The fixed, stable and non-stable pairs• The set of rotations• The graph poset• ρew,f

and ρpw,f

Insight Centre for Data Analytics May 2, 2018 Slide 26

Lemmas

• LetM be a stable matching and S its closed subset

• Let 〈wi , fj〉 be a stable pair

1. If 〈wi , fj〉 ∈ M0, then 〈wi , fj〉 ∈ M iff ρei,j /∈ S .

2. Else, if 〈wi , fj〉 ∈ Mz , then 〈wi , fj〉 ∈ M iff ρpi,j ∈ S .

3. Otherwise, 〈wi , fj〉 ∈ M iff ρpi,j ∈ S ∧ ρei,j /∈ S .

Insight Centre for Data Analytics May 2, 2018 Slide 27

Lemmas

• LetM be a stable matching and S its closed subset

• Let 〈wi , fj〉 be a stable pair

1. If 〈wi , fj〉 ∈ M0, then 〈wi , fj〉 ∈ M iff ρei,j /∈ S .

2. Else, if 〈wi , fj〉 ∈ Mz , then 〈wi , fj〉 ∈ M iff ρpi,j ∈ S .

3. Otherwise, 〈wi , fj〉 ∈ M iff ρpi,j ∈ S ∧ ρei,j /∈ S .

Insight Centre for Data Analytics May 2, 2018 Slide 27

Lemmas

• LetM be a stable matching and S its closed subset

• Let 〈wi , fj〉 be a stable pair

1. If 〈wi , fj〉 ∈ M0, then 〈wi , fj〉 ∈ M iff ρei,j /∈ S .

2. Else, if 〈wi , fj〉 ∈ Mz , then 〈wi , fj〉 ∈ M iff ρpi,j ∈ S .

3. Otherwise, 〈wi , fj〉 ∈ M iff ρpi,j ∈ S ∧ ρei,j /∈ S .

Insight Centre for Data Analytics May 2, 2018 Slide 27

Lemmas

• LetM be a stable matching and S its closed subset

• Let 〈wi , fj〉 be a stable pair

1. If 〈wi , fj〉 ∈ M0, then 〈wi , fj〉 ∈ M iff ρei,j /∈ S .

2. Else, if 〈wi , fj〉 ∈ Mz , then 〈wi , fj〉 ∈ M iff ρpi,j ∈ S .

3. Otherwise, 〈wi , fj〉 ∈ M iff ρpi,j ∈ S ∧ ρei,j /∈ S .

Insight Centre for Data Analytics May 2, 2018 Slide 27

Lemmas

• LetM be a stable matching and S its closed subset

• Let 〈wi , fj〉 be a stable pair

1. If 〈wi , fj〉 ∈ M0, then 〈wi , fj〉 ∈ M iff ρei,j /∈ S .

2. Else, if 〈wi , fj〉 ∈ Mz , then 〈wi , fj〉 ∈ M iff ρpi,j ∈ S .

3. Otherwise, 〈wi , fj〉 ∈ M iff ρpi,j ∈ S ∧ ρei,j /∈ S .

Insight Centre for Data Analytics May 2, 2018 Slide 27

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1

• ∀〈wi , fj〉:1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j

2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j

3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j

4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Rotation-based (SAT) Formulation

• Variables• A Boolean variable xi,j for every pair 〈wi , fj〉• A Boolean variable rk for every rotation ρk

• Constraints

• Closed Subset: ∀ρ1 ≺≺ ρ2: r2 =⇒ r1• ∀〈wi , fj〉:

1. if 〈wi , fj〉 ∈ FP : xi,j2. Else if 〈wi , fj〉 ∈ NSP : ¬xi,j3. Else if 〈wi , fj〉 ∈ M0, then xi,j == ¬rei,j4. Else, if 〈wi , fj〉 ∈ Mz , then xi,j == rpi,j

5. Otherwise, xi,j == rpi,j ∧ ¬rei,j

• Easily translated in SAT (Γ)

Insight Centre for Data Analytics May 2, 2018 Slide 28

Important Properties of the SAT Formula

• LetM2M(I ,X (M2M)) be the stable matching constraint

• Unit propagation on Γ does not maintain arc consistency

• Theorem: LetD be a domain such that unit propagationis performed without failure on Γ. There exists at least asolution inD that satisfies Γ.

Insight Centre for Data Analytics May 2, 2018 Slide 29

Important Properties of the SAT Formula

• LetM2M(I ,X (M2M)) be the stable matching constraint

• Unit propagation on Γ does not maintain arc consistency

• Theorem: LetD be a domain such that unit propagationis performed without failure on Γ. There exists at least asolution inD that satisfies Γ.

Insight Centre for Data Analytics May 2, 2018 Slide 29

Important Properties of the SAT Formula

• LetM2M(I ,X (M2M)) be the stable matching constraint

• Unit propagation on Γ does not maintain arc consistency

• Theorem: LetD be a domain such that unit propagationis performed without failure on Γ. There exists at least asolution inD that satisfies Γ.

Insight Centre for Data Analytics May 2, 2018 Slide 29

Important Properties of the SAT Formula

• LetM2M(I ,X (M2M)) be the stable matching constraint

• Unit propagation on Γ does not maintain arc consistency

• Theorem: LetD be a domain such that unit propagationis performed without failure on Γ. There exists at least asolution inD that satisfies Γ.

Insight Centre for Data Analytics May 2, 2018 Slide 29

Arc Consistency

Arc Consistency

• Idea: use unit propagation as a support check

• Some assignments already have supports

• O(L2) time

Insight Centre for Data Analytics May 2, 2018 Slide 30

Arc Consistency

Arc Consistency

• Idea: use unit propagation as a support check

• Some assignments already have supports

• O(L2) time

Insight Centre for Data Analytics May 2, 2018 Slide 30

Experimental Protocol

• Models:• fr: SAT-formula• ac: Arc Consistency• bc: State-of-the art propagator [16-stability]

• Lexicographical branching (random, min-max random),activity-based search, impact-based search

• Sex-Equal/Balanced Stable matching

Insight Centre for Data Analytics May 2, 2018 Slide 31

Experimental Protocol

• Models:• fr: SAT-formula• ac: Arc Consistency• bc: State-of-the art propagator [16-stability]

• Lexicographical branching (random, min-max random),activity-based search, impact-based search

• Sex-Equal/Balanced Stable matching

Insight Centre for Data Analytics May 2, 2018 Slide 31

Experimental Protocol

• Models:• fr: SAT-formula• ac: Arc Consistency• bc: State-of-the art propagator [16-stability]

• Lexicographical branching (random, min-max random),activity-based search, impact-based search

• Sex-Equal/Balanced Stable matching

Insight Centre for Data Analytics May 2, 2018 Slide 31

Experimental Protocol

• Models:• fr: SAT-formula• ac: Arc Consistency• bc: State-of-the art propagator [16-stability]

• Lexicographical branching (random, min-max random),activity-based search, impact-based search

• Sex-Equal/Balanced Stable matching

Insight Centre for Data Analytics May 2, 2018 Slide 31

Experimental Protocol

• Models:• fr: SAT-formula• ac: Arc Consistency• bc: State-of-the art propagator [16-stability]

• Lexicographical branching (random, min-max random),activity-based search, impact-based search

• Sex-Equal/Balanced Stable matching

Insight Centre for Data Analytics May 2, 2018 Slide 31

Sex-Equal Stable Matching: Optimality Evaluation

0 0.2 0.4 0.6 0.8

0

200

400

600

800

Optimality ratio

CPU

Tim

e

bc-is-bc-as-

bc-lx-mnbc-lx-rdfr -is-fr -as-

fr -lx-mnfr -lx-rdac-is-ac-as-

ac-lx-mnac-lx-rd

• Clear dominance of the SAT formulation

• Arc Consistency does not pay off

Insight Centre for Data Analytics May 2, 2018 Slide 32

Sex-Equal Stable Matching: Optimality Evaluation

0 0.2 0.4 0.6 0.8

0

200

400

600

800

Optimality ratio

CPU

Tim

e

bc-is-bc-as-

bc-lx-mnbc-lx-rdfr -is-fr -as-

fr -lx-mnfr -lx-rdac-is-ac-as-

ac-lx-mnac-lx-rd

• Clear dominance of the SAT formulation

• Arc Consistency does not pay off

Insight Centre for Data Analytics May 2, 2018 Slide 32

Sex-Equal Stable Matching: Quality of The Solution

0 0.2 0.4 0.6 0.8 1

0

200

400

600

800

Objective ratio

CPU

Tim

e

bc-is-bc-as-

bc-lx-mnbc-lx-rdfr -is-fr -as-

fr -lx-mnfr -lx-rdac-is-ac-as-

ac-lx-mnac-lx-rd

Insight Centre for Data Analytics May 2, 2018 Slide 33

Conclusions & Future Research

Contributions

• Robust solutions

• Modularity & Flexibility of CP to solve hard problems

Future Research

• Other stable matching Problems? E.g., preferencesincluding ties? and 3D stable matching?

• From (1,b)-supermatch to (a,b)-supermatch?

• Algorithmic complexity study for robust solutions?

Insight Centre for Data Analytics May 2, 2018 Slide 34

Conclusions & Future Research

Contributions

• Robust solutions

• Modularity & Flexibility of CP to solve hard problems

Future Research

• Other stable matching Problems? E.g., preferencesincluding ties? and 3D stable matching?

• From (1,b)-supermatch to (a,b)-supermatch?

• Algorithmic complexity study for robust solutions?

Insight Centre for Data Analytics May 2, 2018 Slide 34

Conclusions & Future Research

Contributions

• Robust solutions

• Modularity & Flexibility of CP to solve hard problems

Future Research

• Other stable matching Problems? E.g., preferencesincluding ties? and 3D stable matching?

• From (1,b)-supermatch to (a,b)-supermatch?

• Algorithmic complexity study for robust solutions?

Insight Centre for Data Analytics May 2, 2018 Slide 34

Thank you*Picture taken from The New York Times

Insight Centre for Data Analytics May 2, 2018 Slide 35

References I

Emmanuel Hebrard.Robust solutions for constraint satisfaction and optimisation under uncertainty.PhD thesis, University of New South Wales, 2007.

Vipul Bansal, Aseem Agrawal, and Varun S. Malhotra.Polynomial time algorithm for an optimal stable assignment with multiplepartners.Theor. Comput. Sci., 379(3):317–328, 2007.

Mohamed Siala and Barry O’Sullivan.Revisiting two-sided stability constraints.In Integration of AI and OR Techniques in Constraint Programming - 13thInternational Conference, CPAIOR 2016, Banff, AB, Canada, May 29 - June 1,2016, Proceedings, pages 342–357, 2016.

Dan Gusfield and Robert W. Irving.The Stable Marriage Problem: Structure and Algorithms.MIT Press, Cambridge, MA, USA, 1989.

Insight Centre for Data Analytics May 2, 2018 Slide 36

References II

Begum Genc, Mohamed Siala, Gilles Simonin, and Barry O’Sullivan.On the complexity of robust stable marriage.In Combinatorial Optimization and Applications - 11th International Conference,COCOA 2017, Shanghai, China, December 16-18, 2017, Proceedings, Part II,pages 441–448, 2017.

Mohamed Siala, Emmanuel Hebrard, and Marie-José Huguet.An optimal arc consistency algorithm for a chain of atmost constraints withcardinality.In Principles and Practice of Constraint Programming - 18th InternationalConference, CP 2012, Québec City, QC, Canada, October 8-12, 2012.Proceedings, pages 55–69, 2012.

Mohamed Siala, Christian Artigues, and Emmanuel Hebrard.Two clause learning approaches for disjunctive scheduling.In Principles and Practice of Constraint Programming - 21st InternationalConference, CP 2015, Cork, Ireland, August 31 - September 4, 2015,Proceedings, pages 393–402, 2015.

Insight Centre for Data Analytics May 2, 2018 Slide 37

References III

Mohamed Siala and Barry O’Sullivan.Rotation-based formulation for stable matching.In Principles and Practice of Constraint Programming - 23rd InternationalConference, CP 2017, Melbourne, VIC, Australia, August 28 - September 1,2017, Proceedings, pages 262–277, 2017.

Christian Artigues, Emmanuel Hebrard, Valentin Mayer-Eichberger, MohamedSiala, and Toby Walsh.SAT and hybrid models of the car sequencing problem.In Integration of AI and OR Techniques in Constraint Programming - 11thInternational Conference, CPAIOR 2014, Cork, Ireland, May 19-23, 2014.Proceedings, pages 268–283, 2014.

Mohamed Siala and Barry O’Sullivan.Revisiting two-sided stability constraints.In Integration of AI and OR Techniques in Constraint Programming - 13thInternational Conference, CPAIOR 2016, Banff, AB, Canada, May 29 - June 1,2016, Proceedings, pages 342–357, 2016.

Insight Centre for Data Analytics May 2, 2018 Slide 38

References IV

Emmanuel Hebrard and Mohamed Siala.Explanation-based weighted degree.In Integration of AI and OR Techniques in Constraint Programming - 14thInternational Conference, CPAIOR 2017, Padua, Italy, June 5-8, 2017,Proceedings, pages 167–175, 2017.

Guillaume Escamocher, Mohamed Siala, and Barry O’Sullivan.From backdoor key to backdoor completability: Improving a known measure ofhardness for the satisfiable csp.In Integration of AI and OR Techniques in Constraint Programming - 15thInternational Conference, CPAIOR June 2018, Delft, The Netherlands., 2018.

Mohamed Siala, Emmanuel Hebrard, and Marie-José Huguet.An optimal arc consistency algorithm for a particular case of sequence constraint.

Constraints, 19(1):30–56, 2014.

Nina Narodytska, Thierry Petit, Mohamed Siala, and Toby Walsh.Three generalizations of the FOCUS constraint.Constraints, 21(4):495–532, 2016.

Insight Centre for Data Analytics May 2, 2018 Slide 39

References V

Mohamed Siala, Emmanuel Hebrard, and Marie-José Huguet.A study of constraint programming heuristics for the car-sequencing problem.Engineering Applications of Artificial Intelligence, 38:34–44, 2015.

Matthew L. Ginsberg, Andrew J. Parkes, and Amitabha Roy.Supermodels and robustness.In In AAAI/IAAI, pages 334–339, 1998.

Danuta Sorina Chisca, Mohamed Siala, Gilles Simonin, and Barry O’Sullivan.New models for two variants of popular matching.In 29th IEEE International Conference on Tools with Artificial Intelligence(ICATI) November 2017, Boston, Massachussets, USA.

Nina Narodytska, Thierry Petit, Mohamed Siala, and Toby Walsh.Three generalizations of the FOCUS constraint.In IJCAI 2013, Proceedings of the 23rd International Joint Conference onArtificial Intelligence, Beijing, China, August 3-9, 2013, pages 630–636, 2013.

Begum Genc, Mohamed Siala, Barry O’Sullivan, and Gilles Simonin.Finding robust solutions to stable marriage.In Proceedings of the Twenty-Sixth International Joint Conference on ArtificialIntelligence, IJCAI 2017, Melbourne, Australia, August 19-25, 2017, pages631–637, 2017.

Insight Centre for Data Analytics May 2, 2018 Slide 40