3. Solving CSPs

In order to solve CSPs efficiently, we can continue to iterate on our search methodology.

Backtracking Search

Backtracking Search

Backtracking search is a variation of the Depth First Search in which we do:

• Single-variable assignments Only assignment to one variable at each level of the search tree, this accounts for commutativity: e.g. assigning A to red then B to green is the same as B to green then A to red. This reduces paths from $n{!}^{\ast }{m}^n$ to $m^n$.
• For-each assignment Check for failure. Backtracks on failure to next assignment. If all assignments fail, return failure.

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Backtracking Procedure

Below describes a procedure for recursive backtracking:

def recursiveBacktracking(assignment, csp):
	if assignment.complete:
		return assignment
	
	var = selectUnassignedVariable(variables[csp], assignment, csp)
	for each value in orderDomainValues(var, assignment, csp):
		if value is consistent with assignment given constraints[csp]:
			assignment.add({ var = value })
			result = recursiveBacktracking(assignment, csp)
			if not result.failure:
				return result
			
		assignment.remove({ var = value })
	
	throw "failure"

Further improvements to backtracking search

However this is still not complete and it’s inefficient.

• We are still exhaustively trying all combinations until failure at a given level.
• There are an arbitrary selection of values.

To further improve the algorithm, we could:

• Decide which variable to assign at each level.
• Decide the order in which the values are tried.
• Try and detect failure early.

So what can we do to improve?

To decide which variable to assign next, we could look at:

• Most Constrained Variable

  Most Constrained Variable: choose the variable with the fewest legal values. “Minimum Remaining Values (MRV) heuristic”

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• Most Constraining Variable

  Most Constraining Variable: choose the variable that imposes the most constraints on the remaining values for other variables.

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Once we select a variable, we also need to figure out which value to pick:

Choose the least constraining value: value that rules out the fewest values in the remaining variables.

We can also detect failure early:

• Forward Checking (Search)

  Forward Checking: propagate information from assigned to unassigned variables. We keep track of remaining legal values for unassigned variables and terminate the search when any variable has no legal values.

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• Constraint Propagation

  Constraint Propagation: communicate domain reduction of a decision variable to all of the constraints stated over said variable.

  It is often much faster than traditional state-space search. It can eliminate areas of the state space that are no longer relevant.

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  There are different forms of this such as:

  • Arc Consistency

    Arc Consistency: keep pairs of variables consistent, $X\to Y$ is consistent iff for each value of $X$ there are allowed values of $Y$. When checking $X\to Y$, throw out any values of $X$ for which there isn’t an allowed value of $Y$.

    In this example, the blue value in NSW is thrown out.

    If $X$ loses a value, all pairs $Z\to X$ need to be rechecked. Here we have now run into a situation where we can’t satisfy the problem. So we’ve eliminated the original assignment we’ve chosen.

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Arc Consistency Algorithm (AC-3)

Wikipedia: https://en.wikipedia.org/wiki/AC-3_algorithm Given a CSP, we try to reduce its domains with the following algorithm:

def AC-3(csp):
	queue = csp.arcs
	
	while queue is not empty:
		(i, j) = removeFirst(queue)
		if removeInconsistentValues(i, j):
			for each k in neighbors[i]:
				queue.push((k, i))
 
def removeInconsistentValues(i, j):
	removed = False
	
	for each x in domain[i]:
		if no value y in domain[j] allows (x, y) to satisfy constain i <-> j:
			delete x from domain[i]
			removed = True
	
	return removed

Going back to our backtracking search algorithm, we can implement it like so:

def recursiveBacktracking(assignment, csp):
	if assignment.complete:
		return assignment
	
	var = selectUnassignedVariable(variables[csp], assignment, csp)
	for each value in orderDomainValues(var, assignment, csp):
		if value is consistent with assignment given constraints[csp]:
			assignment.add({ var = value })
			
			# run our inferences, such as AC-3
			inferences = INFERENCE(csp, var, value)
			if inferences do not fail:
				assignment.add(inferences)
				result = recursiveBacktracking(assignment, csp)
				if not result.failure:
					return result
			
		assignment.remove({ var = value })
	
	throw "failure"

Relations

All of the examples we’ve looked at so far rely on unary constraints.

todo: program lion / unicorn example?