Chapter5 Constraint Satisfaction Problems 20070412 Chap5 1 - - PDF document

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20070412 Chap5 1

Chapter5

Constraint Satisfaction Problems

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Constraint Satisfaction Problems (CSPs)

Standard Search Problem

• State is a “black box”, can be represented by an

arbitrary data structure that can be accessed only by the problem-specific routines --- the successor functions, heuristic function, and goal test.

Constraint Satisfaction Problem

• State and goal test conform to a standard, structured,

and very simple representation.

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CSP is defined by

a set of variables, X1, X2, …, Xn, with values from domain D1, D2, …, Dn, and a set of constraints, C1, C2, …, Cm, specifying allowable combinations of values for subsets of variables

State is defined by an assignment of values to some or all of the variables, {Xi = vi , Xj = vj , …}

Constraint Satisfaction Problems

(cont.-1)

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Consistent (or legal) assignment

an assignment that does not violate any constraints.

Complete assignment

• ne in which every variable is mentioned.

Solution

is a complete assignment that satisfies all the constraints.

Some CSPs also require a solution that maximizes an objective function

Constraint Satisfaction Problems

(cont.-2)

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Example: Map-Coloring

Variables: WA, NT, Q, NSW, V, SA, T Domains: Di = {red, green, blue} Constraints: adjacent regions must have different colors e.g. WA ≠ NT or (WA, NT) ∈{(red, green), (red, blue), (green, red), (green, blue), …}

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Example: Map-Coloring (cont.)

Solutions: assignments satisfying all constraints

e.g. {WA = red, NT = green, Q = red, NSW = green, V= red, SA = blue, T = green}

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Constraint Graph

• It is helpful to visualize a CSP as a constraint graph.
• Nodes are variable, and arcs show constraints.
• CSP benefits
• Standard representation pattern
• Generic goal and successor functions
• Generic heuristics (no domain

specific expertise).

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Varieties of CSPs

Discrete variables

• finite domains, size d ⇒ O(dm) complete assignments
• e. g. 8 Queens, Q1, …Q8, with the domain {1, 2, 3, 4, 5, 6, 7, 8}
• infinite domains, (integers, strings, etc.)

need a constraint language (cannot enumerate all allowed combinations of values)

• e. g. job scheduling, Job1 which takes 5 days, must precede Job3

StartJob1 + 5 ≤ StartJob3

Continuous variables

e g. Hubble Space Telescope observations

• Linear constraints are solvable in polynomial time

by Linear programming methods.

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Varieties of Constraints

Unary constraints, involves a single variable

e.g. SA ≠ green

Binary constraints, involves pairs of variables

e.g. SA ≠ WA

Higher-order constraints, involves 3 or more variables

e.g. cryptarithmetic column constraints Every high-order, finite-domain constraint can be reduced to a set of binary constraints if enough auxiliary variables are

• introduced. (Exercise 5.11).

We deal only with binary constraints in this chapter.

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Varieties of Constraints (cont.)

Absolute vs. Preference constraints

Preference constraints can often be encoded as costs on individual variable assignments e.g. red is better than green

• Prof. X might prefer teaching in the morning,

whereas prof. Y prefers teaching in the afternoon.

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Example: Cryptarithmetic

Variables: F T U W R O X1 X2 X3 Domains: { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Constraints: Alldiff(F, T, U, W, R, O) O + O = R + 10 * X1 X1 + W + W = U + 10 * X2 X2 + T + T = O + 10 * X3 X3 = F

X3 X2 X1 Where X1, X2 and X3 are auxiliary variables

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Standard Search Formulation (Incremental)

Initial state: the empty assignment, { } Successor function: assign a value to an

unassigned variable that does not conflict with current assignment

Goal test: the current assignment is complete. Path cost: a constant cost for every step.

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Backtracking Search for CSPs

In all CSPs, variable assignments are commutative.

[WA = red then NT = green] same as [NT = green then WA = red ]

Only need to consider assignments to a single value at each node.

there are dn leaves (where d: domain size, n: number of variables)

Backtracking search

a form of depth-first search for CSP with single–variable assignments

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Backtracking Search for CSPs (cont.)

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Backtracking Example

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Backtracking Example (cont.-1)

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Backtracking Example (cont.-2)

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Backtracking Example (cont.-3)

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Improving backtracking efficiency

Previous improvements introduce heuristics General-purpose methods can give huge gains in speed:

• Which variable should be assigned next?
• In what order should its values be tried?
• Can we detect inevitable failure early?
• Can we take advantage of problem structure?

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Some Key Questions of Backtracking Search

Variable and Value Ordering

Which variable should be assigned next, and in what order should its values be tried?

Propagating information through constraints

What are the implications of the current variable assignments for the other unassigned variables?

Intelligent backtracking

When a path fails --- that is, a state is reached in which a variable has no legal values --- can the search avoid repeating this failure in subsequent paths?

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Minimum Remaining Values

Minimum remaining value (MRV) heuristic

• r Most constrained variable heuristic
• r Fail-first heuristic
• choose the variable with the fewer legal values

var ← SELECT-UNASSIGNED-VARIABLE(VARIABLES[csp],assignment,csp)

After WA = red and NT = green, SA is assigned next rather than assigning Q

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Degree heuristic

Degree heuristic (Most Constraining Variable)

• tie-breaker among most constrained variables
• choose the variable with the most constraints
• n remaining variables

SA (degree is 5) is assigned first

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Least Constraining Value

After WA = red and NT = green, Q is assigned to red Blue is bad choice since it eliminated the last legal value left for Q’s neighbor, SA

• Least constraining value heuristic
• try to leave the maximum flexibility for subsequent variable

assignment

• prefer the value that rules out the fewest choices

for the neighboring variables in the constraint graph

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Constraint Propagation

Propagating the implications of a constraint

• n one variable onto other variables
• Forward checking
• Arc consistency (more stronger)

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Forward Checking

Idea:

Keep track of remaining legal values for unassigned variables Terminate search when any variable has no legal values Whenever a variable X is assigned, the forward checking process looks at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain any value that is inconsistent with the value chosen for X.

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Forward Checking (cont.-1)

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Forward Checking (cont.-2)

After WA = red

NT can no longer be red SA can no longer be red

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Forward Checking (cont.-3)

After WA = red and Q = green With MRV, NT = blue and SA = blue

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Forward Checking (cont.-4)

After WA = red, Q = green and V = blue, leaving SA with no legal values

FC has detected that partial assignment is inconsistent with the constraints and backtracking can occur.

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Forward Checking (cont.-5)

Forward checking propagates information from assigned to unassigned variable, but does not provide early detection for all failures. After WA = red and Q = green With MRV, NT = blue and SA = blue, but they cannot both be blue

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Example: 4-Queens Problem

1 3 2 4 3 2 4 1

X1 { 1,2,3,4} X3 { 1,2,3,4} X4 { 1,2,3,4} X2 { 1,2,3,4} [4-Queens slides copied from B.J. Dorr CMSC 421 course on AI]

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Example: 4-Queens Problem (cont.-1)

1 3 2 4 3 2 4 1

X1 { 1,2,3,4} X3 { 1,2,3,4} X4 { 1,2,3,4} X2 { 1,2,3,4}

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Example: 4-Queens Problem (cont.-2)

1 3 2 4 3 2 4 1

X1 { 1,2,3,4} X3 { ,2, ,4} X4 { ,2,3, } X2 { , ,3,4}

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Example: 4-Queens Problem (cont.-3)

1 3 2 4 3 2 4 1

X1 { 1,2,3,4} X3 { ,2, ,4} X4 { ,2,3, } X2 { , ,3,4}

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Example: 4-Queens Problem (cont.-4)

1 3 2 4 3 2 4 1

X1 { 1,2,3,4} X3 { , , , } X4 { ,2, , } X2 { , ,3,4}

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Example: 4-Queens Problem (cont.-5)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1,2,3,4} X4 { 1,2,3,4} X2 { 1,2,3,4}

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Example: 4-Queens Problem (cont.-6)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, ,3, } X4 { 1, ,3,4} X2 { , , ,4}

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Example: 4-Queens Problem (cont.-7)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, ,3, } X4 { 1, ,3,4} X2 { , , ,4}

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Example: 4-Queens Problem (cont.-8)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, , , } X4 { 1, ,3, } X2 { , , ,4}

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Example: 4-Queens Problem (cont.-9)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, , , } X4 { 1, ,3, } X2 { , , ,4}

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Example: 4-Queens Problem (cont.-10)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, , , } X4 { , ,3, } X2 { , , ,4}

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Example: 4-Queens Problem (cont.-11)

1 3 2 4 3 2 4 1

X1 { ,2,3,4} X3 { 1, , , } X4 { , ,3, } X2 { , , ,4}

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Arc Consistency

Simplest form of propagation makes each arc consistent. X → Y is consistent iff for every value x of X, there is some allowed y Applying arc consistency has result in early detection of an inconsistency that is not detected by pure forward checking.

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Arc Consistency (cont.-1)

SA → NSW is consistent iff SA=blue and NSW=red

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Arc Consistency (cont.-2)

NSW → SA is consistent iff NSW=red and SA=blue NSW=blue and SA=??? Arc can be made consistent by removing blue from NSW

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Arc Consistency (cont.-3)

V → NSW Arc can be made consistent by removing blue from NSW If X loses a value, neighbors of X need to be rechecked. Remove red from V

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Arc Consistency (cont.-4)

SA → NT, but SA becomes empty.

Arc consistency detects failure earlier than forward checking Can be run as a preprocessor or after each assignment Repeated until no inconsistency remains

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Arc Consistency Algorithm

O(n2d3) can be reduces to O(n2d2) But cannot detect all failures in polynomial time.

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k-consistency

A CSP is k-consistent,

if for any k-1 variables and for any consistent assignment to those variables, a consistent value can always be assigned to any kth variable. k=1 i.e. node consistent k=2 i.e. arc consistent k=3 i.e. path consistent

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Strongly k-consistency

A graph is strongly k-consistent if

• It is k-consistent and
• It is also (k-1) consistent, (k-2) consistent, … all the

way down to 1-consistent.

This is ideal since a solution can be found in time O(nd) instead of O(n2d3) YET no free lunch: any algorithm for establishing n-consistency must take time exponential in n, in the worst case.

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Further improvements

Checking special constraints

• Checking Alldiff(…) constraint

If m variables involved, n possible distinct values and m > n then the constraint cannot be satisfied.

e.g. {WA=red, NSW=red} 3 variables (SA, NT and Q) can be only 2 values {green, blue}

• Checking Atmost(…) constraint

Bounds propagation for larger value domains

e.g. atmost(10, PA1, PA2, PA3, PA4)

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Further improvements (cont.-1)

Intelligent backtracking

• Standard form is chronological backtracking

i.e. try different value for preceding variable.

• More intelligent, backtrack to conflict set.

Set of variables that caused the failure or set of previously assigned variables that are connected to X by constraints. Backjumping moves back to most recent element of the conflict set. Forward checking can be used to determine conflict set.

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Local Search for CSPs (Iterative algorithms)

Hill-climbing, simulated annealing typically use complete state representation. (i.e., all variables assigned.) To apply to CSPs

• allow states with unsatisfied constraints
• operators: reassign variable values
• variable selection: randomly select any conflicted variable
• value selection: min-conflict heuristic

Select new value that results in a minimum number of conflicts with other variables.

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Min-Conflict Algorithm

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An An 4 4-

• queens Problem

queens Problem

States: 4 queens in 4 columns (44 = 256 states) Operators: move queen in column Goal test: no attacks Evaluation: h(n) = number of attacks

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An An 4 4-

• queens

queens Problem Problem(cont

(cont.) .)

Assume one queen in each column. Which row does each one go in?

Variables Q1, Q2, Q3, Q4 Domains Di = {1, 2, 3, 4} Constraints Qi ≠ Qj (cannot be in same row) |Qi − Qj| ≠ |i − j| (or same diagonal) Translate each constraint into set of allowable values for its variables E.g., values for (Q1,Q2) are (1, 3) (1, 4) (2, 4) (3, 1) (4, 1) (4, 2)

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An 8 An 8-

• queens Problem

queens Problem

Algorithm chosen: the min-conflicts heuristic repair method Algorithm Characteristics: repairs inconsistencies in the current configuration selects a new value for a variable that results in the minimum number of conflicts with other variables

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Detailed Steps Detailed Steps

• 1. One by one, find out the number of conflicts between

the inconsistent variable and other variables.

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• 2. Choose the one with the smallest number of

conflicts to make a move

Detailed Steps Detailed Steps (cont.

(cont.-

• 1)

1)

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• 3. Repeat previous steps until all the

inconsistent variables have been assigned with a proper value.

Detailed Steps Detailed Steps (cont.

(cont.-

• 2)

2)

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Performance of min-conflicts

• Given random initial state, can solve n-queens in almost

constant time for arbitrary n with high probability (e.g., n = 10,000,000)

• The same appears to be true for any randomly-generated

CSP except in a narrow range of the ratio

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Problem Structure

• How can the problem structure help to find a solution quickly?
• Subproblem identification is important
• Coloring Tasmania and mainland are independent subproblems
• Identifiable as connected components of constrained graph.
• Improves performance

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Problem Structure (cont.)

Completely independent subproblems are delicious, then, but rare.

Suppose each subproblem has c variables out of n total With decomposition solution cost : n/c * dc, --- linear in n Without decomposition solution cost : dn, --- exponential in n

e.g. n=80, d=2, c=20 280 = 4 billion years at 10 million nodes/sec 4 * 220 = 0.4 sec at 10 million nodes/sec

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Tree-Structured CSPs

• In most cases subproblems of a CSP are connected as a tree.
• Any tree-structured CSP can be solved in time linear in the

number of variables.

• Theorem: if the constraint graph has no loops, the CSP can

be solved in O(n d2) time. (Compare to general CSPs, where worse-case time is O(dn)

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Algorithm for Tree-Structured CSPs

1. Choose a variable as root, order variables from root to leaves, such that every node’s parent precedes it in the

• rdering.

2. For j from n down to 2, apply RemoveInconsistent (Parent(Xj), Xj) 3. For j from 1 to n, assign Xj consistently with Parent(Xj)

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Nearly Tree-Structured CSPs

To reduce constraint graph to tree

• by removing nodes
• by collapsing nodes

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Nearly Tree-Structured CSPs (cont.-1)

Idea: assign values to some variables so that the remaining variables form a tree. Assume that we assign {SA=x} cycle cutset

And remove any values from the other variables that are inconsistent. The selected value for SA could be the wrong one so we have to try all of them

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Nearly Tree-Structured CSPs (cont.-2)

• This approach is worthwhile if cycle cutset is small.
• Finding the smallest cycle cutset is NP-hard
• Approximation algorithms exist
• This approach is called cutset conditioning.

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Nearly Tree-Structured CSPs (cont.-3)

• Necessary requirements:

Every variable appears in at least one of the subproblems. If two variables are connected in the original problem, they must appear together in at least one subproblem. If a variable appears in two subproblems, it must appear in every subproblem along the path connecting those subproblems

• Tree decomposition of the

constraint graph in a set of connected subproblems.

• Each subproblem is solved

independently.

• Resulting solutions are

combined