Sep 10, 2003
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- recap arc consistency

- Arc consistency = strong 2-consistency

- What is 1-consistency by the way?
	- remove any violating variable from a domain

- Definitions
	- A graph is K-consistent if:
		- choose values for any (K-1) variables
	          (that satisfy constraints among them)
		- choose a Kth variable, then
		- there exists a value for this variable
	- A graph is strongly K-consistent if:
		- it is 1-consistent,
		- it is 2-consistent,
		- ...
		- it is (K-1)-consistent, and
		- it is K-consistent

- An n-node constraint graph
	- can be made n-consistent
		- in an aim to "read off" the answer
	- but sometimes, just k-consistent is enough
		- for k<n

- Interestingly,
	- as long as k is made > width of graph, then
	  you need no search/backtracking!

- Thus, two ways to solve a CSP
	- solving a CSP using constraint propagation
		- can just 'read off' the answer
	- solving a CSP using search

- In general to just 'read off' the answer
	- need to maintain n-consistency
	  for a graph with n nodes

- Sometimes k<n is sufficient
	- example of a 3 node graph where arc-consistency
	  is enough

- How do we determine k?
	- k should be set just > width of the constraint graph

- Computing the width of the constraint graph
	- first make 'tree-like' decompositions of the graph
	- for each decomposition
		- x = size of largest subproblem
		- y = x -1
	- find the minimum y across all decompositions

- Rules for making a 'tree-like' decomposition
	- every variable must appear in at least one of
	  the subproblems
	- every constraint must appear in at least one of the
	  subproblems
	- if a variable appears in two subproblems, it must
	  also appear in every subproblem 'along' the two
	  subproblems

- Thus far, the two extremes of solving CSPs are
	- constraint propagation (involves a lot of 'thinking')
	- backtracking (might involve thrashing)

- Third approach
	- "mix" CP and search
	- i.e., embed CP in a backtracking algorithm
		- at each node try to achieve *some* consistency
		- AND search

- How to choose orderings 
	- most constrained variable heuristic
	- most constraining variable heuristic (useful for first level)
	- least constraining value heuristic

- SAT problems: a favorite type of CSPs
	- continued in next class