1. Foundations of Constraint Processing
CSCE421/821, Fall 2003:
Berthe Y. Choueiry (Shu-we-ri)
Ferguson Hall, Room 104
Tel: +1(402)
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2. Constraint Satisfaction 101
Motivating example, application areas
Application examples
Definition, representation, & formal characterization
Solving techniques
Issues & research directions
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3. How to solve a CSP? Search 1. Constructive, systematic
2. Iterative repair
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4. Systematic search Starting from a root node
Consider all values for a variable V1
For every value for V1, consider all values for V2
etc..
For n variables, each of domain size d
Maximum depth? fixed!
Maximum number of paths? size of search space, size of CSP
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5. Systematic search Before starting search, consider
Importance of modeling/formulation:
to control the size of the search space
Preprocessing (i.e., constraint filtering/propagation, consistency checking)
to reduce size of search space
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6. Importance of modeling
N-queen: formulation 1
Variables?
Domains?
Size of CSP?
N-queens: formulation 2
4 Column positions
4 Rows V1 V2 V3 V4
16 Cells
V11 V12 V13 V14
V21 V22 V23 V24
V31 V32 V33 V34
V41 V42 V43 V44
{0,1}
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7. Constraint checking Arc-consistency B 1- B: [ 5 .. 14 ] A < B A13
1- B: [ ]
A < B 14 A
[ ]
C: [ ] 2
2- A: [ ]
B < C
[ ]
C: [ ]
2 < C - A < 5
3- B: [ ] 6 14
[ ] C
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8. Constraint checking
Arc-consistency: every combination of two adjacent variables
3-consistency, k-consistency (k  n)
Constraint filtering, constraint checking, etc..
Eliminate non-acceptable tuples prior to search
Warning: arc-consistency does not solve the problem
still is not a solution!
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9. Systematic search (I) Back-checking
Systematic search generates dn possibilities
Are all possibilities acceptable?
Expand a partial solution only when it is consistent
This yields early pruning of inconsistent paths
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10. Systematic search (II) Chronological backtracking
What if only one solution is needed?
Depth-first search & Chronological backtracking
DFS: Soundness? Completeness?
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11. Systematic search (III) Intelligent backtracking
What if the reason for failure was higher up in the tree?
Backtrack to source of conflict !!
Backjumping, conflict-directed backjumping, etc.
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12. Systematic search (IV) Ordering heuristics
Which variable to expand first?
Heuristics:
most constrained variable first (reduce branching factor)
most promising value first (find quickly first solution)
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13. Systematic search (V) Back-checking
Search tree with only backtrack search?
Root node
Backtrack 2 Q Q 1 3 Q 4 Q
Backtrack 1 2 4 Q 1 2 3 2 Q 3
Backtrack 4 1 2 3 4 1 1 Q Q 3
Solution!
26 nodes Visited 1 2 3 4 1 2
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14. Systematic search (VI) Forward checking
Search Tree with domains filter by Forward Check
Root Node Q 1 2
Domain Wipe out V4 Q Q 3
Domain Wipe Out V3 4 Q Q 4 2 Q Q 1 3 Q
Solution!
8 nodes visited
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15. Summary of backtrack search
Constructive, systematic, exhaustive
In general sound and complete
Back-checking: expands nodes consistent with past
Backtracking: Chronological vs. intelligent
Ordering heuristics:
Static
Dynamic variable
Dynamic variable-value pairs
Looking ahead:
Partial look-ahead
Forward checking (FC)
Directional arc-consistency (DAC)
Full (a.k.a. Maintaining Arc-consistency or MAC)
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16. CSP: a decision problem (NP-complete)
Modeling: abstraction and reformulation
Preprocessing techniques:
eliminate non-acceptable tuples prior to search
Systematic search:
potentially dn paths of fixed lengths
chronological backtracking
intelligent backtracking
variable/value ordering heuristics
Search ‘hybrids:’
Mixing consistency checking with search: look-ahead strategies
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17. Non-systematic search
Iterative repair, local search: modifies a global but inconsistent solution to decrease the number of violated constraints
Examples: Hill climbing, taboo search, simulated annealing, GSAT, WalkSAT, Genetic Algorithms, Swarm intelligence, etc.
Features: Incomplete & not sound
Advantage: anytime algorithm
Shortcoming: cannot tell that no solution exists
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18. Decomposition Decomposition Conjunctive Disjunctive
Partition constraints (DB)
Decomposition
Conjunctive
Disjunctive
Properties of disjunctive decompositions
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19. Properties of disjunctive decompositions
Conjunctive or disjunctive?
Consistent: No constraint is removed
Simplifying: Size(Pi) < Size( P)
Semi-complete: At least one solution is kept
Complete: No solution is lost
Redundant: Solutions replicated in { Pi}
Reducible: may be < Size( P)
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20. Deep analysis Uncover particular properties, e.g.
bound the required level of consistency (islands of tractability)
predict ease/difficulty of solving a given instance
Structure, topology of the constraint graph
tree, DAGs, chordal, etc.
Types, semantics of the constraints
subsets of Allen's relations, all-diffs, functional, monotonic, row-convex, linear inequalities, etc.
Order parameter (phase transition)
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21. Multi agents 1. Computational tasks: problem decomposed, replicated
2. Types of agent:
broker, solver, problem, solver+problem
3. Architecture and authority:
hierarchical, egalitarian, priority-based (e.g., vote)
4. Nature of communication:
agent-to-agent, group, broadcast
5. Interaction strategies:
cooperating vs. competing
transparent vs. secretive
negotiation + alliance/coalition formation
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22. Computational tasks
At one end of the spectrum, agents may be involved in solving heterogeneous, distinct and completely independent problems and request other agents to supply specific functionalities for the completion of the individual tasks.
At the other end of the spectrum, the same problem could be replicated and assigned to all agents, which can then share their individual results with other agents incrementally in order to speed up the execution of the global computational task.
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23. Types of agent An agent in the system can be any of the following:
an agent that collects data from the environment and formulates the CSP
a reasoning module with specific computational characteristics (e.g., various search algorithms)
brokers that facilitate matching between a service seeker and a number of service providers (e.g., CORBA brokers).
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24. Agent architecture and how is authority granted
Agents could be organized in a strict hierarchy in which a given agent has full control over the activities of the agents that lie underneath it in the hierarchy. It decides how the lower level agents may cooperate while ensuring coordination with the higher-level agent.
Agents could be in a strictly flat structure competing for services and rewards, either chaotically or according to some strict priority policy, for example, based on voting or time-responsiveness.
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25. Communication environment
Communications among agents may be conducted according to:
a one-to-one schema
multi-cast (i.e., one-to-group), or
broadcast, where all agents in the environment have access to the content of the communicated information.
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26. Type of supported interactions
Agents may be cooperative, pooling their resources and capabilities to achieve a common, global objective, or they could be competitive trying to win rewards and optimize their individual gain.
They could also adopt a midway strategy, dynamically forming coalitions and gathering support to acquire more resources and realize greater gains.
Also, agents may be transparent about their intentions, resources, needs, and constraints or may be secretive, hiding one or the other of their strengths or weaknesses.
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27. Research Directions
Preceding (i.e., search, backtrack, iterative repair, V/V/ordering, consistency checking, decomposition, symmetries & interchangeability, deep analysis)
Evaluation of algorithms
worst-case analysis vs. empirical studies
random problems vs. real-world problems
Cross-fertilization:
DB, SAT & theoretical computer science (TCS), mathematical programming, interval mathematics, logical inference, applications, etc.
Modeling & Reformulation
Multi agents
Distribution and negotiation
decomposition & alliance formation
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28. CSP in a nutshell (I) Definition: P = (V, D, C )
Constraint graph, constraint network
Finite domains
Binary constraints, universal constraints
Examples: map coloring, puzzles, resource allocation, temporal reasoning, product configuration, databases, spreadsheets, graphical layouts, graphical user-interfaces, bioinformatics, etc.
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29. CSP in a nutshell (II) Solution technique: Search constructive
Enhancing search:
constructive
iterative repair
Intelligent backtrack
Variable/value ordering Consistency checking
Hybrid search
 Symmetries
 Decomposition
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30. CSP in a nutshell (III) Deep analysis:  Graph topology
exploit problem structure
Research
 Graph topology
 Constraint semantics
Phase transition
- k-ary constraints, soft constraints
- continuous vs. finite domains
- evaluation of algorithms (empirical)
- cross-fertilization (mathematical program.)
 reformulation and approximation
 architectures (multi-agent, negotiation)
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31. Constraint Logic Programming (CLP)
A merger of
 Constraint solving
 Logic Programming, mostly Horn clauses e.g., Prolog)
Building blocks
Constraint: primitives but also user-defined
cumulative/capacity (linear ineq), MUTEX, cycle, etc.
domain: Booleans, natural/rational/real numbers, finite
Rules (declarative): a statement is a conjunction of constraints and is tested for satisfiability before execution proceeds further
Mechanisms: satisfiability, entailment, delaying constraints
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32. Constraint Processing Techniques
Were you to ask me which programming paradigm is likely to gain most in commercial significance over the next 5 years, I would have to pick Constraint Logic Programming (CLP), even though it is perhaps currently one of the least known and understood. That’s because CLP has the power to tackle those difficult combinational problems encountered for instance in job scheduling, timetabling, and routing which stretch conventional programming techniques beyond their breaking point. Though CLP is still the subject of intensive research, it is already being used by large corporation such as manufacturers Michelin and Dassault, the French railway authority SNCF, airlines Swissair and SAS and Cathay Pacific, and Hong Kong International Terminals, the world’s largest privately-owned container terminal Byte, Dick Powntain
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