CS460 Fall 2013 Lecture 4 Constraint Satisfaction Problems

CSP formulation Simplicity of domain Number of variables Simplicity of constraints

Constraint graph Unary constraints Binary constraints N-ary constraints

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

CSP Complexity NP hard Exponential

CSP and Search DFS, uninformed Why is search approach inefficient? Backtracking search – Fail early

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

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

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)

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

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

Value ordering Least constraining value

Handling bigger problems 1000 queens

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

Problem structure Exploit shape of graph Tree structure – No loops Cutset conditioning

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