New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks Sylvain Bouveret and Michel Lemaître Office National d’Études et de Recherches Aérospatiales (ONERA) Centre National d’Études Spatiales (CNES) Institut de Recherche en Informatique de Toulouse (IRIT) Toulouse – France 20 th International Joint Conference on Artificial Intelligence Hyderabad, January 11, 2007
Modeling the problem Solving the problem Introduction Results Conclusion Fairness in combinatorial problems. . . Many real-world combinatorial problems. . . Nurse rostering problem. Balanced timetables. Fair allocation of airport and airspace resources (to several airlines). Fair share of Earth Observation Satellites. . . . are combinatorial collective decision making problems under admissibility constraints, involving directly or indirectly the concept of fairness . Initial question How can we handle fairness requirements in this kind of constraint satisfaction problems ? New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 2 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness in combinatorial problems. . . Many real-world combinatorial problems. . . Nurse rostering problem. Balanced timetables. Fair allocation of airport and airspace resources (to several airlines). Fair share of Earth Observation Satellites. . . . are combinatorial collective decision making problems under admissibility constraints, involving directly or indirectly the concept of fairness . Initial question How can we handle fairness requirements in this kind of constraint satisfaction problems ? New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 2 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Outline Modeling the problem 1 Constraint Satisfaction Problems The leximin criterion Solving the problem 2 Sort and Conquer Using cardinality combinators A branch-and-bound-like algorithm Using cardinality-minimal critical subsets Implementing the problem 3 New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 3 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Outline Modeling the problem 1 Constraint Satisfaction Problems The leximin criterion Solving the problem 2 Sort and Conquer Using cardinality combinators A branch-and-bound-like algorithm Using cardinality-minimal critical subsets Implementing the problem 3 New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 4 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Constraint networks Constraint network [Montanari, 1974] A constraint network is based on : a set of variables X = { x 1 , . . . , x p } ; a set of domains D = { D x 1 , . . . , D x p } ; a set of constraints C , with, for all c ∈ C : X ( c ) the scope of the constraint, R ( c ) the set of allowed tuples of the constraint. Montanari, U. (1974). Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences , 7:95–132. New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 5 / 31
Modeling the problem Solving the problem Introduction Results Conclusion The Constraint Satisfaction Problem Classical CSP Given : A constraint network ( X , D , C ) . Is there a complete consistent instantiation v of ( X , D , C ) ? ❀ NP -complete. CSP with objective variable Given : A constraint network ( X , D , C ) and an objective variable o ∈ X , such that D o ⊂ N . What is the maximal value α of D o such that there is a complete consistent instantiation b v with b v ( o ) = α ? ❀ NP -complete (decision problem). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 6 / 31
Modeling the problem Solving the problem Introduction Results Conclusion The Constraint Satisfaction Problem Classical CSP Given : A constraint network ( X , D , C ) . Is there a complete consistent instantiation v of ( X , D , C ) ? ❀ NP -complete. CSP with objective variable Given : A constraint network ( X , D , C ) and an objective variable o ∈ X , such that D o ⊂ N . What is the maximal value α of D o such that there is a complete consistent instantiation b v with b v ( o ) = α ? ❀ NP -complete (decision problem). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 6 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness requirements ? Hypotheses: The preferences are numerical. The quality of a solution is measured only on the basis of the agents utilities ( Welfarism ). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 7 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness requirements ? Hypotheses: The preferences are numerical. The quality of a solution is measured only on the basis of the agents utilities ( Welfarism ). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 7 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness requirements ? Hypotheses: The preferences are numerical. The quality of a solution is measured only on the basis of the agents utilities ( Welfarism ). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 7 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness requirements ? Hypotheses: The preferences are numerical. The quality of a solution is measured only on the basis of the agents utilities ( Welfarism ). 7 2 3 4 1 New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 7 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Fairness requirements ? Hypotheses: The preferences are numerical. The quality of a solution is measured only on the basis of the agents utilities ( Welfarism ). 7 2 3 4 1 Utility profile: − → u = � 3 , 4 , 2 , 1 , 7 � New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 7 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. Classical utilitarian SWO [Harsanyi] v ⇔ � n i = 1 u i ≤ � n → − u � − → i = 1 v i . Features The agents are utility “producers”. It is indifferent to inequalities between agents ❀ it can lead to very unfair decisions. New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. Classical utilitarian SWO [Harsanyi] v ⇔ � n i = 1 u i ≤ � n → − u � − → i = 1 v i . Fairness and utilitarian SWO � 10 , 10 , 10 , 10 � � � 41 , 0 , 0 , 0 � , whereas � 10 , 10 , 10 , 10 � is more equitable than � 41 , 0 , 0 , 0 � . New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. Egalitarian SWO [Rawls] → − u � − → v ⇔ min n i = 1 u i ≤ min n i = 1 v i . Features It only takes the least satisfied agent into account ❀ natural inclination to fairness. New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. Egalitarian SWO [Rawls] − → u � − → v ⇔ min n i = 1 u i ≤ min n i = 1 v i . Features It only takes the least satisfied agent into account ❀ natural inclination to fairness. On the other hand, it can lead to non Pareto-optimal decisions (drowning effect). New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
Modeling the problem Solving the problem Introduction Results Conclusion Classical Social Welfare Orderings Classical utilitarian SWO. Egalitarian SWO. Leximin SWO. Egalitarian SWO [Rawls] → − u � − → v ⇔ min n i = 1 u i ≤ min n i = 1 v i . Egalitarian SWO and Pareto-efficiency � 1 , 1 , 1 , 1 � ∼ � 1000 , 1 , 1000 , 1000 � , whereas � 1 , 1 , 1 , 1 � and � 1000 , 1 , 1000 , 1000 � are very different ! New Constraint Programming Approaches For The Computation Of Leximin-Optimal Solutions In Constraint Networks 8 / 31
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