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.
Slide 2
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
Slide 3
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)
Slide 4
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)
Slide 5
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)
Slide 6
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)
Slide 7
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.
Slide 8
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
Slide 9
Generic and specific models Valued CN Semiring CN Soft as
Hard
Slide 10
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
Slide 11
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
Slide 12
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
Slide 13
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
Slide 14
September 2005First international summer school on constraint
processing14 Possibilistic constraint network ( =max) x3x3 x2x2
x5x5 x1x1 x4x4 F(X): highest color used (b