Sep 6, 2006
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- Recall Constraint Graphs

- Primary way of solving CSPs: Backtracking 
	- really dfs
	- possibility of thrashing
		- primarily due to arc-inconsistency
		- but also due to node-inconsistency

- Definition of node-inconsistency

- Defn of arc-consistency and (inconsistency)
	- arc {V_i, V_j} is arc consistent iff
	  for all a in D_i, there exists "a" b in D_j
          such that (a,b) is in R_ij	
	
- Can make a constraint graph arc consistent by 
	- removing values in domain D_i for which there 
          are no compatible values in D_j
		
- An example of constraint propagation to restore
	- arc consistency

- What can happen if arc consistency is enforced?
	- domains are emptied => no solution
	- all domains are of size 1 => one solution
	- all but one domains are of size 1,
          one domain has k values => k solutions
	- other => could be any number of solutions, 
          including none

- Algorithms
	- CLEANUP(Vi,Vj)
		- to make arc arc-consistent
	- Using CLEANUP
		- to create a full-fledged constraint propagation algo.
		- AC-1 (not so efficient)
  	- AC-3
		- only revisit those edges potentially changed by REVISIon
	- If (Vi,Vj) is updated, we only need to visit
	  (Vk,Vi) where k<>i and k<>j
		- why?

- Recall that arc-consistency cannot simplify problems completely
	- still some search might be required

- Some more things we need to complete the big picture
	- interleaving constraint propagation and search
	- how to do CSP search better

- Interleaving search and constraint propagation
	- At each level in the search tree, use the partial assignment
	  of variables to create more constrained CSPs that might be
	  amenable to constraint propagation (e.g., arc-consistency)

- How to search better
	- having decided that some search is inevitable?

- Two useful heuristics
	- most constrained variable
	- least constraining value

- Why do these work?
	- CSPs are commutative in nature
		- order of var assignment doesn't matter