Methods that transform the original network into a tighter and tighter representations

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Preston Summers
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2 pproximation of inference: rc, path and i-consistecy Methods that transform the original network into a tighter and tighter representations all 00 IS 75 - onstraint Networks

3 rc-consistency X, Y, Z, T X < Y Y = Z T < Z X T X,, Y,,,,,, T < = < Z all 00 IS 75 - onstraint Networks

4 rc-consistency X, Y, Z, T X < Y Y = Z T < Z X T X T < Y < Z = all 00 IS 75 - onstraint Networks 4

5 all 00 IS 75 - onstraint Networks 5

6 R RX = {,,}, RY = {,,}, constriant reduces domain of X to R = {,}. R X X < Y all 00 IS 75 - onstraint Networks 6

7 i i π i ( Rij j ) all 00 IS 75 - onstraint Networks 7

8 !" all 00 IS 75 - onstraint Networks 8

9 # omplexity (Mackworth and reuder, 986): e = number of arcs, n variables,k values O ( enk (ek^, each loop, nk number of loops), best-case = ek, rc-consistency is: Ω( ek ) ) all 00 IS 75 - onstraint Networks 9

10 omplexity: O( ek ) est case O(ek), since each arc may be processed in O(k) all 00 IS 75 - onstraint Networks 0

11 $%&"" ' (%( &&" all 00 IS 75 - onstraint Networks

12 ) omplexity: O( ek ) (ounter is the number of supports to ai in xi from xj. S_(xi,ai) is the set of pairs that (xi,ai) supports) all 00 IS 75 - onstraint Networks

13 * && Implement - distributedly. Node x_j sends the message to node x_i i h j i i π π i ( Rij j i ( Rij j ) ) Node x_i updates its domain: i π ( R ) i i ij j = i i h j i Messages can be sent asynchronously or scheduled in a topological order all 00 IS 75 - onstraint Networks

14 +, Example: a triangle graph-coloring with values. Is it arc-consistent? Is it consistent? It is not path, or -consistent. all 00 IS 75 - onstraint Networks 4

15 - all 00 IS 75 - onstraint Networks 5

16 - all 00 IS 75 - onstraint Networks 6

17 R ij omplexity: O(k^) est-case: O(t) Worst-case O(tk) R ij π ij ( Rik k Rkj ) all 00 IS 75 - onstraint Networks 7

18 -# omplexity: O( n 5 k 5 O(n^) triplets, each take O(k^) steps O(n^ k^) Max number of loops: O(n^ k^). ) all 00 IS 75 - onstraint Networks 8

19 -. omplexity: Optimal P-4: O( n k 5 O( n ) k ) (each pair deleted may add: n- triplets, number of pairs: O(n^ k^) size of Q is O(n^ k^), processing is O(k^)) all 00 IS 75 - onstraint Networks 9

20 $%&"& P- requires processings of each arc while P- may not an we do path-consistency distributedly? all 00 IS 75 - onstraint Networks 0

21 + all 00 IS 75 - onstraint Networks

22 /"" ' "" all 00 IS 75 - onstraint Networks

23 omplexity: for binary constraints i or arbitrary constraints: O((k) ) O( k i ) all 00 IS 75 - onstraint Networks

24 )!%&" all 00 IS 75 - onstraint Networks 4

25 + all 00 IS 75 - onstraint Networks 5

26 0"( x x π ( R }) x S S {x omplexity: O(t k), t bounds number of tuples. Relational arc-consistency: R π }( R S { x} S { x S x ) all 00 IS 75 - onstraint Networks 6

27 $%&""( {x+y+z <= 5, z >= } x<=, y<= Example of relational arc-consistency { G, G}, all 00 IS 75 - onstraint Networks 7

28 Global constraints: e.g., all-different constraints Special semantic constraints that appears often in practice and a specialized constraint propagation. Used in constraint programming. ounds-consistency: pruning the boundaries of domains o exercise 6 all 00 IS 75 - onstraint Networks 8

29 $%&""" = {,4,5,6} = {,4} = {,,4,5} = {,,4} E = {,4} lldiff (,,,,E} rc-consistency does nothing pply G to sol(,,,,e)? = {6}, = {}. lg: bipartite matching kn^.5 (Lopez-Ortiz, et. l, IJI-0 pp 45 ( fast and simple algorithm for bounds consistency of alldifferent constraint) all 00 IS 75 - onstraint Networks 9

30 0"" lldifferent Sum constraint Global cardinality constraint (a value can be assigned a bounded number of times) The cummulative constraint (related to scheduling tasks) all 00 IS 75 - onstraint Networks 0

31 all 00 IS 75 - onstraint Networks

32 "" all 00 IS 75 - onstraint Networks

33 "&& ( V ~) and () is arc-consistent relative to but not vice-versa rc-consistency achieved by resolution: res(( V ~),) = Given also ( V ), path-consistency means: res(( V ~),( V )) = ( V ) What can generalized arc-consistency do to cnfs? Relational arc-consistency rule = unit-resolution all 00 IS 75 - onstraint Networks

34 "&& If lex goes, then ecky goes: If hris goes, then lex goes: Query: Is it possible that hris goes to the party but ecky does not? Is propositional theory ϕ = {,,, } (or, ) (or, ) satisfiable? all 00 IS 75 - onstraint Networks 4

35 &&" && all 00 IS 75 - onstraint Networks 5

36 " "" "( Think about the following: G-i apply -i to the dual problem when singleton variables are explicit: the bi-partite representation. What is the complexity? Relational arc-consistency: imitate unit propagation. pply - on the dual problem where each subset of a scope is presented. Is unit propagation equivalent to -4? all 00 IS 75 - onstraint Networks 6

37 x [,0], x + arc by y = 0 adding y [5,5], consistency x + y x [,5], = 0, y y [5,9] 5 z [ 0,0], y + z path consistency x z 7 obtained by adding, x + y = 0, y z all 00 IS 75 - onstraint Networks 7

38 "" all 00 IS 75 - onstraint Networks 8

39 &" '& ) all 00 IS 75 - onstraint Networks 9

40 * && Implement - distributedly. i i π i ( Rij j ) Node x_j sends the message to node x_i h j i π i ( Rij j ) Node x_i updates its domain: i i h j i Generalized arc-consistency can be implemented distributedly: sending messages between constraints over the dual graph: R π }( R S { x} S { x S x ) all 00 IS 75 - onstraint Networks 40

41 all 00 IS 75 - onstraint Networks 4 G R R R 4 R R 5 R 6 *"" %&" G The message that R sends to R is R updates its relation and domains and sends messages to neighbors

42 * R- can be applied to the dual problem of any constraint network. G b) onstraint network all 00 IS 75 - onstraint Networks 4

43 all 00 IS 75 - onstraint Networks 4 G R R R 4 R R 5 R 6 G *"4&

44 all 00 IS 75 - onstraint Networks 44 G +# G h h h h h 4 h 5 h 4 h 5 4 h 4 h 6 4 h 6 h 5 6 h 4 5 h 5 h 5 h 6 R R R 4 R R 5 R 6

45 all 00 IS 75 - onstraint Networks 45 G R R R 4 R R 5 R 6 G #

46 all 00 IS 75 - onstraint Networks 46 G G h h h h h 4 h 5 h 4 h 5 4 h 4 h 6 4 h 6 h 5 6 h 4 5 h 5 h 5 h 6 R R R 4 R R 5 R 6 +.

47 +. R R R R G R 5 R 6 G all 00 IS 75 - onstraint Networks 47

48 h 5 + h 4 h R R h h h 4 R h h 5 4 h 6 4 h 4 h R G R 5 5 h 5 h 5 h 6 R 6 6 h 4 6 h 5 G all 00 IS 75 - onstraint Networks 48

49 + R R R R G R 5 R 6 G all 00 IS 75 - onstraint Networks 49

50 h 5 +) h 4 h R R h h h 4 R h h 5 4 h 6 4 h 4 h R G R 5 5 h 5 h 5 h 6 R 6 G 6 h 4 6 h 5 all 00 IS 75 - onstraint Networks 50

51 +) R R R R G R 5 R 6 G all 00 IS 75 - onstraint Networks 5

52 +5 R h h h 4 h 5 h 4 h R R h h 5 4 h 6 4 h 4 h R G R 5 5 h 5 h 5 h 6 R 6 G 6 h 4 6 h 5 all 00 IS 75 - onstraint Networks 5

53 +5 R R R R G R 5 R 6 G all 00 IS 75 - onstraint Networks 5

Consistency algorithms. Chapter 3

Consistency algorithms. Chapter 3 Consistency algorithms Chapter Fall 00 Consistency methods pproximation of inference: rc, path and i-consistecy Methods that transform the original network into tighter and tighter representations Fall