1. Arc-consistency for alldiff(x,y,...,z) Example: P=(X,D,C) Variables: {w,x,y,z}  X Domains: 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(w  d, x  b, y  a, z  c) = true  value b for w is not supported!

2. alldiff: value graph Variables {w,x,y,z}  X Domains 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 w x y z b c d e a

3. alldiff: value graph Variables {w,x,y,z}  X Domains 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 w x y z b c d e a Given a graph G=(V,E), a matching is M  E such that no two edges of M share a vertex

4. alldiff: value graph Variables {w,x,y,z}  X Domains 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 w x y z b c d e a Consistent assignment = maximal cardinality matching covering all the variables We want to remove edges that do not belong to any max-card matching

5. 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

6. 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

7. 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 S  V if it covers all the vertices of S A vertex v is M-free is M does not cover v f ab c ed

8. 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 f ab c ed

9. Step 1 Finding M-augmenting paths w x y z b c d e a

10. Step 1 Finding M-augmenting paths w x y z b c d e a Direct the graph

11. Step 1 Finding M-augmenting paths is easy for bipartite graphs w x y z b c d e a Find a path from an M-free vertex on the left to an M-free vertex on the right

12. Step 1 Finding M-augmenting paths is easy for bipartite graphs w x y z b c d e a Find a path from an M-free vertex on the left to an M-free vertex on the right

13. Step 1 Finding M-augmenting paths is easy for bipartite graphs w x y z b c d e a Find a path from an M-free vertex on the left to an M-free vertex on the right

14. Step 1 w x y z b c d e a Find a path from an M-free vertex on the left to an M-free vertex on the right n breadth-first traversals O(n e) Can be improved to O(n 0.5 e)

15. Step 2: Compute all even-length M-alternating circuit w x y z b c d e a

16. Step 2: Compute all even-length M-alternating circuit w x y z b c d e a Direct the graph

17. Step 2: Compute all even-length M-alternating circuit w x y z b c d e a Compute all strongly Connected components [Tarjan 72] O(e)

18. 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|)

19. Step 3: Compute all even-length M-alternating paths starting at a M-free vertex w x y z b c d e a

20. Step 3: Compute all even-length M-alternating paths starting at a M-free vertex w x y z b c d e a Direct the graph

21. Step 3: Compute all even-length M-alternating paths starting at a M-free vertex w x y z b c d e a Compute alternating Paths starting from free Vertices Breadth-first O(e)

22. Step 4 w x y z b c d e a Remove Edges not identified in previous steps

23. Step 4 w x y z b c d e a Remove Edges not identified in previous steps

24. Step 4 w x y z b c d e a Remove Edges not identified in previous steps

25. Step 4 w x y z b c d e a Remove Edges not identified in previous steps

26. Step 4 w x y z b c d e a Remove Edges not identified in previous steps

27. knapsack constraint Notation: L ≤ ax+by+...+cz ≤ U L, U, a, b,..., cpositive integer constants x, y,..., z integer variables (posit. domains) Example:  Variables: x,y,v,z  Domains: {0, 1}  10 ≤ 2x + 3y + 4v + 5z ≤ 12 Applications: everywhere

28. knapsack constraint Example:  Variables: x,y,v,z  Domains: {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!!

29. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

30. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

31. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

32. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

33. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph  Value 0 of z can be removed x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

34. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph  Value 0 of z can be removed x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0

35. knapsack: propagation Knapsack graph 10 ≤ 2x + 3y + 4v + 5z ≤ 12 Reduced graph  Value 0 of z can be removed Complexity: O(nU 2 )  n is the number of variables  Pseudo-polynomial x y v z 12 11 10 9 8 7 6 5 4 3 2 1 0