26. Constraint Satisfaction Problems: Path Consistency

(1)

Foundations of Artificial Intelligence

26. Constraint Satisfaction Problems: Path Consistency

Malte Helmert

University of Basel

April 19, 2021

M. Helmert (University of Basel) Foundations of Artificial Intelligence April 19, 2021 1 / 15

Foundations of Artificial Intelligence

April 19, 2021 — 26. Constraint Satisfaction Problems: Path Consistency

26.1 Beyond Arc Consistency 26.2 Path Consistency

26.3 Summary

Constraint Satisfaction Problems: Overview

Chapter overview: constraint satisfaction problems:

I 22.–23. Introduction I 24.–26. Basic Algorithms

I 24. Backtracking I 25. Arc Consistency I 26. Path Consistency I 27.–28. Problem Structure

26. Constraint Satisfaction Problems: Path Consistency Beyond Arc Consistency

26.1 Beyond Arc Consistency

(2)

Beyond Arc Consistency: Path Consistency

idea of arc consistency:

I For every assignment to a variable u

there must be a suitable assignment to every other variable v . I If not: remove values of u for which

no suitable “partner” assignment to v exists.

tighter unary constraint on u

This idea can be extended to three variables (path consistency):

I For every joint assignment to variables u, v

there must be a suitable assignment to every third variable w . I If not: remove pairs of values of u and v for which

no suitable “partner” assignment to w exists.

tighter binary constraint on u and v German: Pfadkonsistenz

Beyond Arc Consistency: i -Consistency

general concept of i -consistency for i ≥ 2:

I For every joint assignment to variables v ₁ , . . . , v _i ₋₁

there must be a suitable assignment to every i-th variable v _i . I If not: remove value tuples of v ₁ , . . . , v _i−1 for which

no suitable “partner” assignment for v _i exists.

tighter (i − 1)-ary constraint on v ₁ , . . . , v i−1

I 2-consistency = arc consistency I 3-consistency = path consistency (*) We do not consider general i-consistency further as larger values than i = 3 are rarely used

and we restrict ourselves to binary constraints in this course.

(*) usual definitions of 3-consistency vs. path consistency differ

(*)

when ternary constraints are allowed

26. Constraint Satisfaction Problems: Path Consistency Path Consistency

26.2 Path Consistency

Path Consistency: Definition

Definition (path consistent)

Let C = hV , dom, (R _uv )i be a constraint network.

(a)

Two different variables u, v ∈ V are path consistent with respect to a third variable w ∈ V if

for all values d _u ∈ dom(u), d _v ∈ dom(v ) with hd _u , d _v i ∈ R _uv there is a value d _w ∈ dom(w ) with hd _u , d _w i ∈ R _uw and hd _v , d _w i ∈ R _vw .

(b)

The constraint network C is path consistent if for any three variables u, v , w ,

the variables u and v are path consistent with respect to w .

(3)

Path Consistency: Remarks

remarks:

I Even if the constraint R _uv is trivial, path consistency can infer nontrivial constraints between u and v . I name “path consistency”:

path u → w → v leads to new information on u → v

w

u v

Path Consistency: Example

red blue

v ₁

red blue

v ₂

red blue

v ₃

6= 6=

6=

arc consistent, but not path consistent

Processing Variable Triples: revise-3

analogous to revise for arc consistency:

function revise-3(C, u, v , w):

hV , dom, (R _uv )i := C for each hd _u , d _v i ∈ R _uv :

if there is no d _w ∈ dom(w ) with hd _u , d _w i ∈ R _uw and hd _v , d _w i ∈ R _vw :

remove hd _u , d _v i from R _uv

input: constraint network C and three variables u, v , w of C effect: u, v path consistent with respect to w .

All violating pairs are removed from R _uv .

time complexity: O(k ³ ) where k is maximal domain size

Enforcing Path Consistency: PC-2

analogous to AC-3 for arc consistency:

function PC-2(C):

hV , dom, (R _uv )i := C queue := ∅

for each set of two variables {u, v }:

for each w ∈ V \ {u, v }:

insert hu, v , w i into queue while queue 6= ∅:

remove any element hu, v , w i from queue revise-3(C, u, v , w )

if R _uv changed in the call to revise-3:

for each w ⁰ ∈ V \ {u, v }:

insert hw ⁰ , u, v i into queue insert hw ⁰ , v , ui into queue

(4)

PC-2: Discussion

The comments for AC-3 hold analogously.

I PC-2 enforces path consistency

I proof idea: invariant of the while loop:

if hu, v , w i ∈ / queue, then u, v path consistent with respect to w

I time complexity O(n ³ k ⁵ ) for n variables and maximal domain size k (Why?)

26. Constraint Satisfaction Problems: Path Consistency Summary

26.3 Summary

26. Constraint Satisfaction Problems: Path Consistency Summary

Summary

I generalization of

arc consistency (considers pairs of variables) to path consistency (considers triples of variables)

and i-consistency (considers i-tuples of variables)

I arc consistency tightens unary constraints I path consistency tightens binary constraints I i-consistency tightens (i − 1)-ary constraints I higher levels of consistency more powerful

but more expensive than arc consistency

26. Constraint Satisfaction Problems: Path Consistency

Foundations of Artificial Intelligence

26. Constraint Satisfaction Problems: Path Consistency

Malte Helmert

University of Basel

April 19, 2021

Foundations of Artificial Intelligence

April 19, 2021 — 26. Constraint Satisfaction Problems: Path Consistency

26.1 Beyond Arc Consistency 26.2 Path Consistency

26.3 Summary

Constraint Satisfaction Problems: Overview

Chapter overview: constraint satisfaction problems:

I 22.–23. Introduction I 24.–26. Basic Algorithms

I 24. Backtracking I 25. Arc Consistency I 26. Path Consistency I 27.–28. Problem Structure

26.1 Beyond Arc Consistency

Beyond Arc Consistency: Path Consistency

idea of arc consistency:

I For every assignment to a variable u

there must be a suitable assignment to every other variable v . I If not: remove values of u for which

no suitable “partner” assignment to v exists.

tighter unary constraint on u

This idea can be extended to three variables (path consistency):

I For every joint assignment to variables u, v

there must be a suitable assignment to every third variable w . I If not: remove pairs of values of u and v for which

no suitable “partner” assignment to w exists.

tighter binary constraint on u and v German: Pfadkonsistenz

Beyond Arc Consistency: i -Consistency

general concept of i -consistency for i ≥ 2:

I For every joint assignment to variables v 1 , . . . , v i −1

there must be a suitable assignment to every i-th variable v i . I If not: remove value tuples of v 1 , . . . , v i−1 for which

no suitable “partner” assignment for v i exists.

tighter (i − 1)-ary constraint on v 1 , . . . , v i−1

I 2-consistency = arc consistency I 3-consistency = path consistency (*) We do not consider general i-consistency further as larger values than i = 3 are rarely used

and we restrict ourselves to binary constraints in this course.

(*) usual definitions of 3-consistency vs. path consistency differ

(*)

when ternary constraints are allowed

26.2 Path Consistency

Path Consistency: Definition

Definition (path consistent)

Let C = hV , dom, (R uv )i be a constraint network.

Two different variables u, v ∈ V are path consistent with respect to a third variable w ∈ V if

for all values d u ∈ dom(u), d v ∈ dom(v ) with hd u , d v i ∈ R uv there is a value d w ∈ dom(w ) with hd u , d w i ∈ R uw and hd v , d w i ∈ R vw .

The constraint network C is path consistent if for any three variables u, v , w ,

the variables u and v are path consistent with respect to w .

Path Consistency: Remarks

remarks:

I Even if the constraint R uv is trivial, path consistency can infer nontrivial constraints between u and v . I name “path consistency”:

path u → w → v leads to new information on u → v

w

u v

Path Consistency: Example

red blue

v 1

red blue

v 2

red blue

v 3

6= 6=

6=

arc consistent, but not path consistent

Processing Variable Triples: revise-3

analogous to revise for arc consistency:

function revise-3(C, u, v , w):

hV , dom, (R uv )i := C for each hd u , d v i ∈ R uv :

if there is no d w ∈ dom(w ) with hd u , d w i ∈ R uw and hd v , d w i ∈ R vw :

remove hd u , d v i from R uv

input: constraint network C and three variables u, v , w of C effect: u, v path consistent with respect to w .

All violating pairs are removed from R uv .

time complexity: O(k 3 ) where k is maximal domain size

Enforcing Path Consistency: PC-2

analogous to AC-3 for arc consistency:

function PC-2(C):

hV , dom, (R uv )i := C queue := ∅

for each set of two variables {u, v }:

for each w ∈ V \ {u, v }:

insert hu, v , w i into queue while queue 6= ∅:

remove any element hu, v , w i from queue revise-3(C, u, v , w )

if R uv changed in the call to revise-3:

for each w 0 ∈ V \ {u, v }:

I For every joint assignment to variables v ₁ , . . . , v _i ₋₁

there must be a suitable assignment to every i-th variable v _i . I If not: remove value tuples of v ₁ , . . . , v _i−1 for which

no suitable “partner” assignment for v _i exists.

tighter (i − 1)-ary constraint on v ₁ , . . . , v i−1

Let C = hV , dom, (R _uv )i be a constraint network.

for all values d _u ∈ dom(u), d _v ∈ dom(v ) with hd _u , d _v i ∈ R _uv there is a value d _w ∈ dom(w ) with hd _u , d _w i ∈ R _uw and hd _v , d _w i ∈ R _vw .

I Even if the constraint R _uv is trivial, path consistency can infer nontrivial constraints between u and v . I name “path consistency”:

v ₁

v ₂

v ₃

hV , dom, (R _uv )i := C for each hd _u , d _v i ∈ R _uv :

if there is no d _w ∈ dom(w ) with hd _u , d _w i ∈ R _uw and hd _v , d _w i ∈ R _vw :

remove hd _u , d _v i from R _uv

All violating pairs are removed from R _uv .

time complexity: O(k ³ ) where k is maximal domain size

hV , dom, (R _uv )i := C queue := ∅

if R _uv changed in the call to revise-3:

for each w ⁰ ∈ V \ {u, v }:

insert hw ⁰ , u, v i into queue insert hw ⁰ , v , ui into queue

I time complexity O(n ³ k ⁵ ) for n variables and maximal domain size k (Why?)