Constraint Satisfaction R N Chapter 5 Constraint Satisfaction

Constraint Satisfaction R & N Chapter 5

Constraint Satisfaction – Early Use From http: //www-2. cs. cmu. edu/~awm/731/constraint 07 -2 x 2. pdf

The Waltz Algorithm

More on Waltz Algorithm The constraints come from the fact that a line must have the same label at both ends.

Crossword Puzzles

Latin Squares Using only the numbers 1, 2, 3, and 4, arrange four sets of these numbers into a four-by-four array so that no column or row contains the same two numbers. The result is known as a Latin square. Here are two examples of Latin squares of order 4: 1 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 2 1 4 3 http: //www. sciencenews. org/20000506/mathtrek. asp

Satisfiability (P Q S) (A Q B) (P B C) ( A B C) (Q A C) Example applications:

Real Examples Scheduling What else?

Algorithms for CSPs • Branching State space: usually incremental formulation • Straight line State space: usually complete-state formulation

Cryptarithmetic SEND +MORE MONEY PEAS description: State space when formulated as a CSP: variables, domain, constraints • Incremental state space formulation: • Complete state formulation:

Cryptarithmetic Representing the constraints: As formulas: As a graph: Using special constraints: Alldiff SEND +MORE MONEY

Cryptarithmetic – Best First Search M=1 M=2 SEND +MORE MONEY

Cryptarithmetic – Best First Search M=1 M=2 M=1 S=2 O =3 … SEND +MORE MONEY

Cryptarithmetic – Branching Algorithms • Backtracking • Minimum remaining values • Degree heuristic • Least-constraining value M=1 M=2 SEND +MORE MONEY

Cryptarithmetic – Constraint Propagation Forward checking: After assigning a value to a variable X, look at all the variables connected to X and prune inconsistent values. Constraint propagation using arc consistency: The tradeoff: how much time to spend propagating vs. searching. SEND +MORE MONEY

Backtracking – Chronological vs. Dependency Directed

Straight-Line Algorithms: Min-Conflicts function Min-Conflicts(csp, max_steps) returns a solution or failure inputs: csp, a constraint satisfaction problem max_steps, the number of steps allowed before giving up current an initial assignment for csp for i = 1 to max_steps do if current is a solution for csp then return current var a randomly chosen, conflicted variable from Variables[csp] value the value v for var that minimizes Conflicts(var, v, current, csp) set var = value in current return failure

Do Straight-Line Algorithms Work? Does Min-Conflicts work? From: http: //www. cs. mu. oz. au/303/slides/week 03 ah. pdf

Do Straight-Line Algorithms Work?

Another Win of Straight Line Algorithms They can be used to perturb a solution that ceases to be correct: Most scheduling problems