AINN ICT4101 FS

Constraint Satisfaction

Problems

Outline

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Constraint Satisfaction Problems (CSP)

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Backtracking search for CSPs

Constraint satisfaction problems (CSPs)

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Standard search problem:

n  state is a "black box“ – any data structure that supports successor

function, heuristic function, and goal test

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CSP:

n  state is defined by variables X_i with values from domain D_i

n  goal test is a set of constraints specifying allowable combinations of

values for subsets of variables

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Simple example of a

formal representation language

Example: Map-Coloring

n  Variables WA, NT, Q, NSW, V, SA, T

n  Domains D_i = {red,green,blue}

n  Constraints: adjacent regions must have different colors

n  e.g., WA ≠ NT, or (WA,NT) in {(red,green),(red,blue),(green,red),

Example: Map-Coloring

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Solutions

are

complete

and

consistent

assignments,

Constraint graph

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Binary CSP:

each constraint relates two variables

Varieties of CSPs

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Discrete variables

n  finite domains:

n  n variables, domain size d à O(dn) complete assignments

n  e.g., Boolean CSPs, incl.~Boolean satisfiability (NP-complete)

n  infinite domains:

n  integers, strings, etc.

n  e.g., job scheduling, variables are start/end days for each job

n  need a constraint language, e.g., StartJob₁ + 5 ≤ StartJob₃

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Continuous variables

n  e.g., start/end times for Hubble Space Telescope observations

Varieties of constraints

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Unary

constraints involve a single variable,

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e.g., SA

≠

green

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Binary

constraints involve pairs of variables,

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e.g., SA

≠

WA

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Higher-order

constraints involve 3 or more

variables,

Example: Cryptarithmetic

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Variables

:

F T U W

R O X

X

X

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Domains

: {

0,1,2,3,4,5,6,7,8,9

}

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Constraints

:

Alldiff (F,T,U,W,R,O)

n  O + O = R + 10 · X₁

Real-world CSPs

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Assignment problems

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e.g., who teaches what class

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Timetabling problems

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e.g., which class is offered when and where?

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Transportation scheduling

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Factory scheduling

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Notice that many world problems involve

Standard search formulation (incremental)

Let's start with the straightforward approach, then fix it

States are defined by the values assigned so far

n  Initial state: the empty assignment { }

n  Successor function: assign a value to an unassigned variable that does

not conflict with current assignment

à fail if no legal assignments

n  Goal test: the current assignment is complete

1.  This is the same for all CSPs

2.  Every solution appears at depth n with n variables

à use depth-first search

3.  Path is irrelevant, so can also use complete-state formulation

Backtracking search

n  Variable assignments are commutative}, i.e.,

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

n  Only need to consider assignments to a single variable at each node

à b = d and there are $d^n$ leaves

n  Depth-first search for CSPs with single-variable assignments is called

backtracking search

n  Backtracking search is the basic uninformed algorithm for CSPs

Improving backtracking efficiency

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General-purpose

methods can give huge

gains in speed:

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Which variable should be assigned next?

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In what order should its values be tried?

Most constrained variable

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Most constrained variable:

choose the variable with the fewest legal values

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a.k.a.

minimum remaining values (MRV)

Most constraining variable

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Tie-breaker among most constrained

variables

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Most constraining variable:

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choose the variable with the most constraints on

Least constraining value

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Given a variable, choose the least

constraining value:

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the one that rules out the fewest values in the

remaining variables

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Combining these heuristics makes 1000

Forward checking

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Idea

:

Forward checking

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Idea

:

Forward checking

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Idea

:

Forward checking

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Idea

:

Constraint propagation

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Forward checking propagates information from assigned to

unassigned variables, but doesn't provide early detection for

all failures:

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NT and SA cannot both be blue!

Arc consistency

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Simplest form of propagation makes each arc

consistent

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X

à

Y

is consistent iff

Arc consistency

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Simplest form of propagation makes each arc

consistent

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X

à

Y

is consistent iff

Arc consistency

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Simplest form of propagation makes each arc

consistent

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X

à

Y

is consistent iff

for every value x of X there is some allowed y

Arc consistency

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Simplest form of propagation makes each arc

consistent

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X

à

Y

is consistent iff

for every value x of X there is some allowed y

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If

X

loses a value, neighbors of

X

need to be rechecked

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Arc consistency detects failure earlier than forward checking

Arc consistency algorithm AC-3

Local search for CSPs

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Hill-climbing, simulated annealing typically work with

"complete" states, i.e., all variables assigned

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To apply to CSPs:

n  allow states with unsatisfied constraints n  operators reassign variable values

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Variable selection: randomly select any conflicted variable

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Value selection by

min-conflicts

heuristic:

n  choose value that violates the fewest constraints

Example: 4-Queens

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States

: 4 queens in 4 columns (4

= 256 states)

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Actions

: move queen in column

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Goal test

: no attacks

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Evaluation

:

h(n)

= number of attacks

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Given random initial state, can solve

n

-queens in almost

Summary

n  CSPs are a special kind of problem:

n  states defined by values of a fixed set of variables

n  goal test defined by constraints on variable values

n  Backtracking = depth-first search with one variable assigned per node

n  Variable ordering and value selection heuristics help significantly

n  Forward checking prevents assignments that guarantee later failure

n  Constraint propagation (e.g., arc consistency) does additional work to

constrain values and detect inconsistencies