Arc-consistency for alldiff(x,y,...,z)

Example: P=(X,D,C)Variables: {w,x,y,z}XDomains:

w{b,c,d,e} x{b,c}y{a,b,c,d} z{b,c}

Constraints: alldiff(w,x,y,z)C

value d for w is supported: alldiff(wd, xb, y a, zc) = true

value b for w is not supported!

alldiff: value graph

Variables {w,x,y,z}XDomains D(w)={b,c,d,e}, D(x)={b,c},

D(y)={a,b,c,d}, D(z)={b,c}Constraint alldiff(w,x,y,z)C

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alldiff: value graph

Variables {w,x,y,z}XDomains D(w)={b,c,d,e}, D(x)={b,c},

D(y)={a,b,c,d}, D(z)={b,c}Constraint alldiff(w,x,y,z)C

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Given a graph G=(V,E), a matching is ME such that no two edges of M share a vertex

alldiff: value graph

Variables {w,x,y,z}XDomains D(w)={b,c,d,e}, D(x)={b,c},

D(y)={a,b,c,d}, D(z)={b,c}Constraint alldiff(w,x,y,z)C

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Consistent assignment =maximal cardinality matchingcovering all the variables

We want to remove edgesthat do not belong to any max-card matching

AC characterization

Theorem: Let G be a graph and M a maximum-cardinality matching. An edge e belongs to some max-card matching iff e is in M, or e is on an even-length M-alternating circuit e is on an even-length M-alternating path

starting at a M-free vertex

AC Algorithm

1. Compute a maximum-cardinality matching.

2. Compute all even-length M-alternating circuits

3. Compute all even-length M-alternating paths starting at a M-free vertex

4. Remove all edges not identified in the previous 3 steps

Step 1. Find Max-card matching

A matching M covers a vertex v if v belongs to some edge in M

A matching M covers a set SV if it covers all the vertices of S

A vertex v is M-free is M does not cover v

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Paths A path is M-alternating if its edges are are

alternatingly out of and in M A M-alternating path is M-augmenting if

has odd length, Its ends are not covered by M

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Step 1

Finding M-augmenting paths

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Step 1

Finding M-augmenting paths

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a Direct the graph

Step 1

Finding M-augmenting paths is easy for bipartite graphs

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a Find a path from anM-free vertex on theleft to an M-free vertex on the right

Step 1

Finding M-augmenting paths is easy for bipartite graphs

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a Find a path from anM-free vertex on theleft to an M-free vertex on the right

Step 1

Finding M-augmenting paths is easy for bipartite graphs

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a Find a path from anM-free vertex on theleft to an M-free vertex on the right

Step 1

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a Find a path from anM-free vertex on theleft to an M-free vertex on the right

n breadth-first traversalsO(n e)

Can be improved toO(n0.5 e)

Step 2:

Compute all even-length M-alternating circuit

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Step 2:

Compute all even-length M-alternating circuit

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a Direct the graph

Step 2:

Compute all even-length M-alternating circuit

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a Compute all stronglyConnected components[Tarjan 72] O(e)

Tarjan (Kosaraju’s)

1. Do a DFS

2. Reverse the direction of edges

3. Do a DFS starting with the vertex visited last in the original DFS

4. Each tree in the forest returned is a SCC

O(|V|+|A|)

Step 3:

Compute all even-length M-alternating paths starting at a M-free vertex

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Step 3:

Compute all even-length M-alternating paths starting at a M-free vertex

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a Direct the graph

Step 3:

Compute all even-length M-alternating paths starting at a M-free vertex

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a Compute alternating Paths starting from free Vertices

Breadth-first O(e)

Step 4

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Remove Edges not identified in previous steps

Step 4

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Remove Edges not identified in previous steps

Step 4

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Remove Edges not identified in previous steps

Step 4

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Remove Edges not identified in previous steps

Step 4

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Remove Edges not identified in previous steps

knapsack constraint

Notation: L ≤ ax+by+...+cz ≤ UL, U, a, b, ..., c positive integer constantsx, y,..., z integer variables (posit.

domains) Example:

Variables: x,y,v,zDomains: {0, 1}10 ≤ 2x + 3y + 4v + 5z ≤ 12

Applications: everywhere

knapsack constraint

Example: Variables: x,y,v,zDomains: {0, 1}10 ≤ 2x + 3y + 4v + 5z ≤ 12

Value 0 of x is supported (0+3+4+5=12) Value 1 of x is supported (2+0+4+5=11) Value 0 of z is not supported!!

knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12

x y v z

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph

Value 0 of z can be removed

x y v z

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph

Value 0 of z can be removed

x y v z

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knapsack: propagation Knapsack graph10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph

Value 0 of z can be removed Complexity: O(nU2)

n is the number of variables Pseudo-polynomial

x y v z

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