Outline What is a Constraint ? What is a Global Constraint ? - - PDF document

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I nterest of Global Constraints for Modeling and Solving

Eric Bourreau, François Laburthe

BOUYGUES e lab St-Quentin-en-Yvelines - FRANCE { ebourrea;flaburth} @challenger.bouygues.fr

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Outline

• What is a Constraint ?
• What is a Global Constraint ?
• Examples of Global Constraints families
• Interest
• Conclusion

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Constraints Principle

• X in 1..10, Y in 1..10, X > Y + 3

– X in 5..10, Y in 1..6 – if min of Y or max of X change, adjust other variable

• X <> Y + 5

– wait until X or Y fixed to a value to remove one value from the domain variation of the other variable

Properties: additive, declarative, partial, multi-directed, data-driven Theoretical frame CSP (constraint satisfaction problem):

Models : unary, binary, … ternary

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Search

• Stating and Solving [Lauriere76]
• Satisfaction (consistent)
• Pro-active : discover infeasible solutions

(inconsistencies) => and react !

• Filtering algorithms
• Saturation
• Implicit enumeration

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Global Constraints

(global vision)

n-ary constraints : alldifferent(X1,X2,X3,X4,X5)

X2 ∈ {1,2,3,4} X3 ∈ {3,4}

≠ ≠

X4 ∈ {3,4} X5 ∈ {1,3,5} X1 ∈ {1,2,3}

Clique of inequalities

X2 X1 X3 X4

2 3 4

X5

5 1

Matching variables ↔ values

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Filtering Algorithms

• Régin 94 :

use strongly connected components in the residual graph

… Régin 96 (global cardinality constraint) … Régin 99 (weighted gcc)

SCS 1 SCS 3 SCS 2 X2 X1 X3 X4

2 3 4

X5

5 1 … Régin 00 (gsc)

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The Right Level

Combinatorial Optimization Problems

Linear Programming

E f f i c i e n c y i n a l g

• r

i t h m s M

• d

e l i n g f

• r

m a l i s m l e v e l s

Constraint Programming

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Global Constraint Family (1)

• Path [Ilog96]
• Cycles covering [Pesant98]
• Circuits partition [Bourreau99]
• Trees cover [Laporte98]

Graph paths

example of propagation in the cycle constraint

Matching, strongly connected components, shortest path

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Global Constraint Family (2)

Scheduling

• Cumulative [Beldiceanu93, Le Pape94]
• Disjunctive [Caseau95, Colombani96]
• Preemptive [Le Pape96]
• Fully Elastic [Baptiste97]
• Trapeze [Poder00]

1 2 3 4 5 6 7 1 2 3 4 5 6 ressource

Tasks interval, edge finding, energetic reasoning

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Global Constraint Family (3)

• Diffn

[Aggoun93] [Beldiceanu-Contejean94]

Computational geometry, Latin squares reasoning, analytical decomposition

Packing, Non overlapping

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Interest : good algorithms

• Adapted algorithms

– Incremental in time – Incremental in memory

• Undirected algorithms (propagation in every

direction)

Q: How to compute Strongly Connected Components in an incremental manner when one edge is remove ? … one node ? … … with minimal complexity in time ? … … with an adapted data structure in memory ? 12

New algorithms … ?

• Necessary conditions to propagate
• Revisiting algorithms

Example :

incremental reduction tests for the Steiner problem in graphs [Duin92, Rottembourg93]… not yet

• Implicit enumeration by exploiting

sub-structures (implicit of explicit)

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High Level Constraints

• Encapsulation

(transparent for users)

• Easy coding

(concise)

• Modeling Tools

(not only solving tools)

• What ≠ how

1 industrial application = 3 sub-problems = 3 constraints 14

Algorithmic Combination

• Multiple visions

Packing

Scheduling

(necessary conditions) as redundant constraints

• Implicit hybridization
• combination of incremental algorithms
• complex use of consistency rules
• efficient and adapted data structures
• dynamic awake of algorithms

4,93

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Caveat

• Time consuming
• nodes explored vs. time spent
• interesting for hard problems
• Not (yet) unified
• example : one cumulative by framework

(Chip, Ilog, Eclipse, Sicstus, Oz, SchedEns, Choco-Igloo, …)

• lack of formalism

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Conclusion

• Where are we today ?
• Setup to collaborate with other paradigms

(automatic conversion in LP, combination with Local Search, bound reinforcement for column generation… )

• Already successful in various areas

coming

Unified theory

arrived

Classification

mature

Techniques