CS 371: Introduction to Artificial Intelligence

Constraint Satisfaction

Outline

Definition of a Constraint Satisfaction Problem

Constraint satisfaction as tree search

Looking ahead: constraint propagation

Looking back: intelligent backtracking

Constraint Optimization Problems

Constraint optimization as a stochastic local search

Introductory Problems

Map Coloring

(see New England + New York map)

Object: to color the map with three colors

N-Queens Problem

Constraint Satisfaction Problem

Variables (X1, X2, …, Xn)

Variable domains (D1, D2, …, Dn)

Variable constraints, pairs of:

Constraint variable subsets, and

Relations over such subsets

Constraint Satisfaction as Tree Search

Initial state:

No variables assigned

Operators:

Assign a single variable a member of its domain

Goal Test:

All variables assigned

Path Cost:

Irrelevant à 0

Issues with Backtracking (DFS) Approaches to CSPs

Large number of variables

Large variable domains

Computational cost of evaluating constraints

Must search entire tree to prove that constraints cannot be satisfied

Reducing Search

Variable/value ordering:

Prefer more constrained variables:

Instantiate (assign) the variable with the fewest possible remaining alternatives

Prefer more constraining variables:

Instantiate the variable which is in the most constraint subsets.

Identify and instantiate a stable set of variables (no direct constraints between pairs) last

Reducing Search (continued)

Looking forward:

Consistency preprocessing:

Constraint propagation: As variables are instantiated, compute some level of impact on future variable assignments

Looking backward: intelligent backtracking

Rather than doing simple DFS backtracking, backtrack to the cause of the constraint violation.

Consistency enforcement

In the simplest case, we try to instantiate a variable with each value in its domain.

What of unary constraints?

e.g. “Vermont must be colored green.”, “Massachusetts must be blue or green.”

Much search can be wasted if such simple constraints are ignored.

Node Consistency

Node consistent – each domain obeys unary constraints

(e.g. Vermont’s domain is reduced to the singleton set “green”.)

Easy to do, avoids wasted work

Usually done at problem formulation level

Suppose New York is colored green. How to propagate this to the constraints of adjacent states?

Arc Consistency

Arc consistent

Node consistent, and

" variables Xi, Xj

" instantiations of Xi,

$ an instantiation of Xj obeying binary constraints between Xi, Xj

Example: If NY green, domains have only values respecting unary constraints, and lacking green for adjacent states.

K-Consistency

The concept of consistency can be generalized.

K-Consistent

" variables X1, X2, …, XK

" instantiations of X1, X2, …X(K-1),

$ an instantiation of XK obeying constraints between XK and {X1, …, X(K-1)}

Strongly K-Consistent

J-Consistent for all J£K

K-Consistency (cont.)

Node Consistent = (Strongly) 1-Consistent

O(de) – d variable domain size, e binary constraints (arcs in constraint graph)

Arc Consistent = Strongly 2-Consistent

O(de²)

Path Consistent = Strongly 3-Consistent

There exists a K for each binary CSP such that enforcing strong K-consistency eliminates backtracking (no search).

Consistency enforcement expensive for non-small K à tradeoff

Constraint Propagation

Different levels of consistency can be enforced during backtracking search.

Common usage (backtracking with arc consistency):

Variable Xi instantiated

For each Xj such that there’s a binary constraint on (Xi, Xj), update Dj

With each domain update, propagate binary constraints onward until no change

Intelligent Backtracking

Example: Maine must be colored green. We start by coloring New Hampshire green, and finish with Maine as the last variable for instantiation. à thrashing

During search, one can keep track of the way in which variable instantiations constrain other variable instantiations.

Then one can backtrack to the cause of instantiation failure.

Combining Approaches

One can combine:

Variable/value ordering heuristics,

Constraint propagation, and

Intelligent backtracking

to greatly increase efficiency of search.

However, tree search is not the only kind of search that can be applied…

Constraint Satisfaction as Iterative Improvement Search

Constraint satisfaction can also be approached using iterative improvement search… (been there, done that already!)

(Constraint Optimization treated in homework project. More slides will be developed on COPs in the future.)


	Definition of a Constraint Satisfaction Problem
	Constraint satisfaction as tree search
	Looking ahead: constraint propagation
	Looking back: intelligent backtracking
	Constraint Optimization Problems
	Constraint optimization as a stochastic local search


	Map Coloring
		(see New England + New York map)
		Object: to color the map with three colors
	N-Queens Problem


	Variables (X1, X2, …, Xn)
	Variable domains (D1, D2, …, Dn)
	Variable constraints, pairs of:
		Constraint variable subsets, and
		Relations over such subsets


	Initial state:
		No variables assigned
	Operators:
		Assign a single variable a member of its domain
	Goal Test:
		All variables assigned
	Path Cost:
		Irrelevant à 0


	Large number of variables
	Large variable domains
	Computational cost of evaluating constraints
	Must search entire tree to prove that constraints cannot be satisfied


Variable/value ordering:
	Prefer more constrained variables:
		Instantiate (assign) the variable with the fewest possible remaining alternatives
	Prefer more constraining variables:
		Instantiate the variable which is in the most constraint subsets.
	Identify and instantiate a stable set of variables (no direct constraints between pairs) last


	Looking forward:
		Consistency preprocessing:
		Constraint propagation: As variables are instantiated, compute some level of impact on future variable assignments
	Looking backward: intelligent backtracking
		Rather than doing simple DFS backtracking, backtrack to the cause of the constraint violation.


	In the simplest case, we try to instantiate a variable with each value in its domain.
	What of unary constraints?
		e.g. “Vermont must be colored green.”, “Massachusetts must be blue or green.”
	Much search can be wasted if such simple constraints are ignored.


	Node consistent – each domain obeys unary constraints
		(e.g. Vermont’s domain is reduced to the singleton set “green”.)
		Easy to do, avoids wasted work
		Usually done at problem formulation level
	Suppose New York is colored green. How to propagate this to the constraints of adjacent states?


Arc consistent
	Node consistent, and
	" variables Xi, Xj
		" instantiations of Xi,
			$ an instantiation of Xj obeying binary constraints between Xi, Xj
Example: If NY green, domains have only values respecting unary constraints, and lacking green for adjacent states.


The concept of consistency can be generalized.
K-Consistent
	" variables X1, X2, …, XK
		" instantiations of X1, X2, …X(K-1),
			$ an instantiation of XK obeying constraints between XK and {X1, …, X(K-1)}
Strongly K-Consistent
	J-Consistent for all J£K


	Node Consistent = (Strongly) 1-Consistent
		O(de) – d variable domain size, e binary constraints (arcs in constraint graph)
	Arc Consistent = Strongly 2-Consistent
		O(de²)
	Path Consistent = Strongly 3-Consistent
	There exists a K for each binary CSP such that enforcing strong K-consistency eliminates backtracking (no search).
	Consistency enforcement expensive for non-small K à tradeoff


	Different levels of consistency can be enforced during backtracking search.
	Common usage (backtracking with arc consistency):
		Variable Xi instantiated
		For each Xj such that there’s a binary constraint on (Xi, Xj), update Dj
		With each domain update, propagate binary constraints onward until no change


	Example: Maine must be colored green. We start by coloring New Hampshire green, and finish with Maine as the last variable for instantiation. à thrashing
	During search, one can keep track of the way in which variable instantiations constrain other variable instantiations.
	Then one can backtrack to the cause of instantiation failure.


	One can combine:
		Variable/value ordering heuristics,
		Constraint propagation, and
		Intelligent backtracking
		to greatly increase efficiency of search.
	However, tree search is not the only kind of search that can be applied…


	Constraint satisfaction can also be approached using iterative improvement search… (been there, done that already!)
	(Constraint Optimization treated in homework project. More slides will be developed on COPs in the future.)