Constraint Satisfaction Problems



Subset of search


Find values for a set of variables, subject to constraints


Variables can be discrete or continuous

Key terms



Assignment


  • Consistent/Legal Assignment
  • Partial Assignment
  • Complete Assignment


Solution


Constraint Types



Unary


Binary


N-ary

Consider Sudoku


Let's write this as a CSP

Recall N Queens


Let's write this as a CSP

Think about Class Scheduling


Think about Map Coloring


Graph-Coloring/Map-Coloring



How can we solve these?

Consider this graph


Try BFS/DFS

Consider this graph


Try BFS/DFS

Better Ordering



Min Remaining Values (vertex)


Degree (vertex)


Least Constraining Value (var value)


Forward Checking / Look Ahead

Constraint Propagation


Choices for one variable limit choices for the next ...


Local consistency →\rightarrow→ Global consistency


Consider Binary constraints

Constraint Propagation

Node Consistency


Every value in domain satisfies unary constraints



E.g.,

X={x}X=\{x\}X={x}

D(x)=Z+D(x) = \mathbb{Z^+}D(x)=Z+

C={(x<8),(x≥5)}C=\{(x < 8), (x\ge 5)\}C={(x<8),(x≥5)}


The restricted domain D(x)={5,6,7}D(x)=\{5,6,7\}D(x)={5,6,7} makes xxx node-consistent

Constraint Propagation

Arc/Edge Consistency


Applies to binary constraints and defined over pairs of variables


∀ x∈D(x),∃ y∈D(y)\forall\ x\in D(x), \exists\ y\in D(y)∀ x∈D(x),∃ y∈D(y) such that (x,y)(x,y)(x,y) satisfy all binary constraints over xxx and yyy

(Every value in domain of xxx has a legal value in domain of yyy)



A CSP is arc-consistent if all pairs of variables are arc-consistent

Constraint Propagation

Arc/Edge Consistency


(Every value in domain of xxx has a legal value in domain of yyy)


E.g.,

X={x,y}X=\{x, y\}X={x,y}

D(x)=D(y)=Z[0,9]D(x) = D(y) = \mathbb{Z}_{[0,9]}D(x)=D(y)=Z[0,9]​

C={(y=x2)}C=\{(y = x^2)\}C={(y=x2)}


The following restricted domain set makes xxx arc-consistent wrt yyy

D(x)={0,1,2,3}D(x) = \{0,1,2,3\}D(x)={0,1,2,3}

D(y)={0,1,4,9}D(y) = \{0,1,4,9\}D(y)={0,1,4,9}

Constraint Propagation

Arc/Edge Consistency


Is arc-consistency symmetric?


X={x,y}X=\{x, y\}X={x,y}

D(x)={2,3,4,5},D(y)={1,3,4,5,6}D(x) = \{2,3,4,5\}, D(y) = \{1,3,4,5,6\}D(x)={2,3,4,5},D(y)={1,3,4,5,6}

C={(x<y)}C=\{(x < y)\}C={(x<y)}


∀ x∈D(x),∃ y∈D(y);  x<y\forall\ x\in D(x), \exists\ y\in D(y);\; x < y∀ x∈D(x),∃ y∈D(y);x<y

However...

∄ x∈D(x) ∀ y∈D(y);  x<y\nexists\ x\in D(x)\ \forall\ y\in D(y);\; x < y∄ x∈D(x) ∀ y∈D(y);x<y

Constraint Propagation

Path Consistency


A 2-variable set (x,y)(x, y)(x,y) is path-consistent with a third variable, zzz if


∀ valid assignments (x=a,y=b),∃ (z=c)\forall\ \textrm{valid assignments } (x=a, y=b), \exists\ (z=c)∀ valid assignments (x=a,y=b),∃ (z=c)
that satisfies constraints on (x,z)(x, z)(x,z) and (z,y)(z, y)(z,y)

Constraint Propagation

Path Consistency


∀ valid assignments (x=a,y=b),∃ (z=c)\forall\ \textrm{valid assignments } (x=a, y=b), \exists\ (z=c)∀ valid assignments (x=a,y=b),∃ (z=c)
that satisfies constraints on (x,z)(x, z)(x,z) and (z,y)(z, y)(z,y)

Constraint Propagation

Path Consistency


∀ valid assignments (x=a,y=b),∃ (z=c)\forall\ \textrm{valid assignments } (x=a, y=b), \exists\ (z=c)∀ valid assignments (x=a,y=b),∃ (z=c)
that satisfies constraints on (x,z)(x, z)(x,z) and (z,y)(z, y)(z,y)

Constraint Propagation

kkk-Consistency


For every valid assignment to a set of k−1k-1k−1 variables,
there exists a valid assignment to the kthk^{th}kth variable


Does not imply k−1k-1k−1 consistency


x<3x < 3x<3

x<yx < yx<y

x+y=zx + y = zx+y=z


D(x)={2,4}D(x) = \{2, 4\}D(x)={2,4}

D(y)={1,3}D(y) = \{1, 3\}D(y)={1,3}

D(z)={3,4,5}D(z) = \{3, 4, 5\}D(z)={3,4,5}