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Soft constraint processing Thomas Schiex INRA Toulouse France Javier Larrosa UPC Barcelona Spain Thanks to School organizers Francesca, Michela, Kryzstof These slides are provided as a teaching support for the community. They can be freely modified and used as far as the original authors (T. Schiex and J. Larrosa) contribution is clearly mentionned and visible and that any modification is acknowledged by the author of the modification.

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September 2005First international summer school on constraint processing2 Overview Introduction and definitions Why soft constraints and dedicated algorithms? Generic and specific models Handling soft constraint problems Fundamental operations on soft CN Solving by search (systematic and local) Complete and incomplete inference Hybrid (search+inference) Polynomial classes Ressources

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September 2005First international summer school on constraint processing3 Why soft constraints? CSP framework: for decision problems Many problems are overconstrained or optimization problems Economics (combinatorial auctions) Given a set G of goods and a set B of bids Bid (B i,V i ), B i requested goods, V i value find the best subset of compatible bids Best = maximize revenue (sum)

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September 2005First international summer school on constraint processing4 Why soft constraints? Satellite scheduling Spot 5 is an earth observation satellite It has 3 on-board cameras Given a set of requested pictures (of different importance) Resources, data-bus bandwidth, setup-times, orbiting select a subset of compatible pictures with max. importance (sum)

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September 2005First international summer school on constraint processing5 Why soft constraints? Probabilistic inference (bayesian nets) Given a probability distribution defined by a DAG of conditional probability tables And some evidence find the most probable explanation for the evidence (product)

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September 2005First international summer school on constraint processing6 Why soft constraints? Ressource allocation (frequency assignment) Given a telecommunication network find the best frequency for each communication link avoiding interferences Best can be: Minimize the maximum frequency (max) Minimize the global interference (sum)

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September 2005First international summer school on constraint processing7 Why soft constraints Even in decision problems: the problem may be unfeasible users may have preferences among solutions It happens in most real problems. Experiment: give users a few solutions and they will find reasons to prefer some of them.

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September 2005First international summer school on constraint processing8 Notations and definitions X={x 1,..., x n } variables ( n variables) D={D 1,..., D n } finite domains (max size d ) YX (Y) = x i Y D i t(Y) = t Y ZY t Y [Z] = projection on Z t Y [-x i ] = t Y [Y-{x i }] Relation R (Y), scope Y, denoted R Y Projection extended to relations

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Generic and specific models Valued CN Semiring CN Soft as Hard

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September 2005First international summer school on constraint processing10 Combined local preferences Constraints are local cost functions Costs combine with a dedicated operator max: priorities +: additive costs *: factorized probabilities Goal: find an assignment with the best combined cost Fuzzy/possibilistic CN Weighted CN Probabilistic CN, BN

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September 2005First international summer school on constraint processing11 Soft constraint network (CN) (X,D,C)(X,D,C) X={x 1,..., x n } variables D={D 1,..., D n } finite domains C={f,...} e cost functions f S, f ij, f i f scope S,{x i,x j },{x i }, f S (t): E (ordered by , T ) Obj. Function: F(X)= f S (X[S]) Solution: F(t) T Soft CN: find minimal solution identity annihilator commutative associative monotonic

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September 2005First international summer school on constraint processing12 Specific frameworks Lexicographic CN, probabilistic CN Instance E TT Classic CN {t,f}{t,f} and t f Possibilistic [0,1] max 0101 Fuzzy CN [0,1] usual min 1010 Weighted CN [0,k]+ 0k0k Bayes net [0,1] 1010

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September 2005First international summer school on constraint processing13 Weighted CSP example ( = +) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): number of non blue vertices For each vertex For each edge: xixi f(x i ) b0 g1 r1 xixi xjxj f(x i,x j ) bbT bg0 br0 gb0 ggT gr0 rb0 rg0 rrT

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September 2005First international summer school on constraint processing14 Possibilistic constraint network ( =max) x3x3 x2x2 x5x5 x1x1 x4x4 F(X): highest color used (b