Interval constraint programming Constraint propagation Trends and open issues Recent solvers Intervals in constraint programming : some trends and open issues Djamila Sam-Haroud Thanks to Gilles Trombettoni for material and comments Swiss Federal institute of technology, School of core computing, EPFL June 2009 Sam-Haroud Intervals in constraint programming
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Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
Intervals in constraint programming :some trends and open issues
Djamila Sam-HaroudThanks to Gilles Trombettoni for material and comments
Swiss Federal institute of technology, School of core computing, EPFL
June 2009
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
3 Trends and open issuesThe dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
4 Recent solvers
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
HC4-reviseBox-reviseLimitations
Contraction using constraint progagation
Contraction/filtering + propagation
The most widely used contraction/filtering tools handle oneconstraint at a time to infer the values to be filtered outfrom the domains
☞ A box is contracted with respect to each individualconstraint
When several constraints are involved, the intersectionbetween their boxes is obtained by an incremental constraintpropagation algorithm (AC3 like algorithms).
(Typical) contraction procedures (Revise) :
HC4-Revise ;
Box-Revise (BoxNarrow) ;Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
3 Trends and open issuesThe dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
4 Recent solvers
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
HC4-reviseBox-reviseLimitations
The Box-Revise procedure
Principle :
Consider the univariate interval constraintc : [f ](x , [y1], ..., [yk ]) = 0 : where every variable, exceptfor x , is replaced by their current domains (intervals).
Reduce [x ] by computing the smallest (l) and the largest(r) zero of c.
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
HC4-reviseBox-reviseLimitations
The Box-Revise algorithm
Split [x ] into sub-intervals [xi ] of width ǫ (processed from left toright in the basic version).
If 0 /∈ [f ]([xi ], [y1], ..., [yk ]), then [xi ] is eliminated and the nextsub-interval considered.
If univariate Interval Newton (interval analysis) certifies a uniquesolution, a zero is identified (at the bounds).
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
HC4-reviseBox-reviseLimitations
Comparison of Revise procedures
When the box computed by a revise procedure is optimal (as tight aspossible with respect to one constraint), it satisfies the2B/Hull-consistency property. This property is hard to achieve
☞ when a constraint contains multiple occurences of the samevariable, the box is overestimated
In practice :
HC4-Revise may not compute the tightest box even if theconstraint contains no multiple occurence of the same variable.
Box-Revise is more costly but better than HC4-Revise when onlyone variable occurs several times in the constraint.
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
3 Trends and open issuesThe dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
4 Recent solvers
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
Intervals are great . . . but
Alternative representations of the constraints exists that cancontribute to contraction more efficiently :
parallelepiped,
zonotopes,
linear relaxations,
convex polyhedral enclosures,
affine arithmetic forms,
Bernstein polynomials,
. . .
Moreover, several different representations can be used cooperativelyduring a single solving process.
☞ Key issue: Decoupling contraction from constraintrepresentation
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
Decoupling contraction from constraint representation
Makes it possible to use cooperatively several constraints
representation. The output may take more accurate forms.Examples :
“Contractor Programming” (Chabert and Jaulin, AI journal, 2008)
☞ proposes a neat paradigm to decouple contraction fromconstraint representation
“Enhancing numerical constraint propagation using multipleinclusion representations” (Vu et al., Annals of mathematics and AI,2009)
☞ generalizes the concepts related to interval forms toprovide a common view of different kind of constraintsrepresentation
☞ proposes a generic combination schema to use severalrepresentations cooperatively
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
Mutliple inclusion representations
Different inclusion representations (interval arithmetic, affinearithmetic ...) can be used to infer redundant constraintsets (PCS: pruning constraint system) to improve contraction :
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
Cooperative use of several representation
Example : combining interval arithmetic, affine arithmetic, and safe linearprogramming enhance contraction (CIRD[ai] algorithm)
☞ showed significant improvements on several test suites
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
3 Trends and open issuesThe dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
4 Recent solvers
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues
Recent solvers
The dependency problemThe nature of constraints mattersIntervals are great . . . butOther sources of improvement
Many research directions
Further developping the algorithms for computing higherdegrees of consistency (like 3B consistency).
Characterization of new global constraints useful forcontraction.
Developing smart splitting and search strategies.
Integration of algebraic approach andcontinuation/homotopy methods.
Developing smart strategies for the cooperative use ofdifferent contraction tools.
Using clustering techniques to reduce the verbosity of theoutput, or to restructure the output with respect toparticular queries (for example connectedness).
. . .
Sam-Haroud Intervals in constraint programming
Interval constraint programmingConstraint propagationTrends and open issues