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Exploring Constraint Satisfaction Problems: Concepts, Techniques, and Examples

This lecture explores Constraint Satisfaction Problems (CSP), focusing on their formulation, complexity, and various techniques for solving them. Key topics include the simplicity of domains and constraints, various types of constraints (unary, binary, n-ary), and examples like the N-Queens problem and Sudoku. The lecture delves into search strategies, including backtracking and arc consistency, and discusses variable and value ordering to enhance efficiency. It also touches upon the NP-hard nature of CSPs and the impacts of problem structure on solving strategies.

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Exploring Constraint Satisfaction Problems: Concepts, Techniques, and Examples

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  1. CS460Fall 2013Lecture 4 Constraint Satisfaction Problems

  2. CSP formulation • Simplicity of domain • Number of variables • Simplicity of constraints

  3. Constraint graph • Unary constraints • Binary constraints • N-ary constraints

  4. CSP Examples • N queens • Map coloring • Cryptarithmetic • Sudoku • Missionaries and Cannibals? • Waltz 3-D shape recognizer • Boolean n-satisfiability problem • Time tabling • Hardware configuration • Layouts

  5. CSP Complexity • NP hard • Exponential

  6. CSP and Search • DFS, uninformed • Why is search approach inefficient? • Backtracking search • Fail early

  7. Backtracking design • Variable order • Failure on violation • Domain value assignment order • Filtering • Exploit graph structure • …

  8. Filtering: forward checking • Filter out choices from domains • Backtrack when choices run out

  9. Constraint propagation • Enforcing arc consistency • “arc” belongs to constraint propagation, not to constraint graph as such • Consistency: For every value in the tail X, there is some Y in the head which can be assigned without violating a constraint • Intuitively, there will be no backtracking generated due to that particular pair of nodes • Use arc consistency to filter out values from domain • “delete from the tail” • Any time a variable domain shrinks, all its neighbors pointing to it have to be re-tested • Visit all the arcs over and over again till every arc is simultaneously consistent • Backtracking, but less intense • Compare to A* search (precomputing heuristic)

  10. Arc consistency: limitations • Does not detect all failures, only a subclass of them • Part of overall backtracking search • Arc consistency has to be enforced after every assignment in a backtracking search

  11. Variable ordering • Pick variable with min remaining values • Fail early (at top of search tree)

  12. Value ordering • Least constraining value

  13. Handling bigger problems • 1000 queens

  14. K-consistency • Generalized arc consistency • Tail is now k-1 consistent subset of nodes. Head is the remaining node. • Strong k-consistency: all lower-orders are also consistent • Lots of pre-computation • Analogy to A* search

  15. Problem structure • Exploit shape of graph • Tree structure • No loops • Cutset conditioning

  16. Local Search

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