1 Consistency algorithms Chapter 3

Spring 2007 ICS 275A - Constraint Networks 2 Consistency methods Approximation of inference: Arc, path and i-consistecy Methods that transform the original network into a tighter and tighter representations

Spring 2007 ICS 275A - Constraint Networks 3 Arc-consistency 32,1, 32,1,32,1, 1  X, Y, Z, T  3 X  Y Y = Z T  Z X  T XY TZ 32,1,  =  

Spring 2007 ICS 275A - Constraint Networks 4 1  X, Y, Z, T  3 X  Y Y = Z T  Z X  T XY TZ  =   Arc-consistency

Spring 2007 ICS 275A - Constraint Networks 5 Arc-consistency

Spring 2007 ICS 275A - Constraint Networks 6 Arc-consistency Only domain constraints are recorded: Example:

Spring 2007 ICS 275A - Constraint Networks 7 Revise for arc-consistency

Spring 2007 ICS 275A - Constraint Networks 8 Figure 3.3: (a) Matching diagram describing a network of constraints that is not arc-consistent (b) An arc-consistent equivalent network.

Spring 2007 ICS 275A - Constraint Networks 9 AC-1 Complexity ( Mackworth and Freuder, 1986 ): e = number of arcs, n variables,k values (ek^2, each loop, nk number of loops), best-case = ek, Arc-consistency is:

Spring 2007 ICS 275A - Constraint Networks 10 AC-3 Complexity: Best case O(ek), since each arc may be processed in O(2k)

Spring 2007 ICS 275A - Constraint Networks 11 Example: A three variable network, with two constraints: z divides x and z divides y (a) before and (b) after AC-3 is applied.

Spring 2007 ICS 275A - Constraint Networks 12 AC-4 Complexity: (Counter is the number of supports to ai in xi from xj. S_(xi,ai) is the set of pairs that (xi,ai) supports)

Spring 2007 ICS 275A - Constraint Networks 13 Example applying AC-4

Spring 2007 ICS 275A - Constraint Networks 14 Distributed arc-consistency (Constraint propagation) Implement AC-1 distributedly. Node x_j sends the message to node x_i Node x_i updates its domain: Messages can be sent asynchronously or scheduled in a topological order

Spring 2007 ICS 275A - Constraint Networks 15 Exercise: make the following network arc-consistent Draw the network’s primal and dual constraint graph Network = Domains {1,2,3,4} Constraints: y < x, z < y, t < z, f<t, x<=t+1, Y<f+2

Spring 2007 ICS 275A - Constraint Networks 16 Arc-consistency Algorithms AC-1: brute-force, distributed AC-3, queue-based AC-4, context-based, optimal AC-5,6,7,…. Good in special cases Important: applied at every node of search (n number of variables, e=#constraints, k=domain size) Mackworth and Freuder (1977,1983), Mohr and Anderson, (1985)…

Spring 2007 ICS 275A - Constraint Networks 17 Is arc-consistency enough? Example: a triangle graph-coloring with 2 values. Is it arc-consistent? Is it consistent? It is not path, or 3-consistent.

Spring 2007 ICS 275A - Constraint Networks 18 Path-consistency

Spring 2007 ICS 275A - Constraint Networks 19 Path-consistency

Spring 2007 ICS 275A - Constraint Networks 20 Revise-3 Complexity: O(k^3) Best-case: O(t) Worst-case O(tk)

Spring 2007 ICS 275A - Constraint Networks 21 PC-1 Complexity: O(n^3) triplets, each take O(k^3) steps  O(n^3 k^3) Max number of loops: O(n^2 k^2).

Spring 2007 ICS 275A - Constraint Networks 22 PC-2 Complexity: Optimal PC-4: (each pair deleted may add: 2n-1 triplets, number of pairs: O(n^2 k^2)  size of Q is O(n^3 k^2), processing is O(k^3))

Spring 2007 ICS 275A - Constraint Networks 23 Example: before and after path- consistency PC-1 requires 2 processings of each arc while PC-2 may not Can we do path-consistency distributedly?

Spring 2007 ICS 275A - Constraint Networks 24 Path-definition of path- consistency

Spring 2007 ICS 275A - Constraint Networks 25 Path-consistency Algorithms Apply Revise-3 (O(k^3)) until no change Path-consistency (3-consistency) adds binary constraints. PC-1: PC-2: PC-4 optimal:

Spring 2007 ICS 275A - Constraint Networks 26 I-consistency

Spring 2007 ICS 275A - Constraint Networks 27 Higher levels of consistency, global-consistency

Spring 2007 ICS 275A - Constraint Networks 28 Revise-i Complexity: for binary constraints For arbitrary constraints:

Spring 2007 ICS 275A - Constraint Networks 29 4-queen example

Spring 2007 ICS 275A - Constraint Networks 30 I-consistency

Spring 2007 ICS 275A - Constraint Networks 31 Path-consistency vs 3- consistency

Spring 2007 ICS 275A - Constraint Networks 32 Arc-consistency for non-binary constraints: Generalized arc-consistency Complexity: O(t k), t bounds number of tuples. Relational arc-consistency:

Spring 2007 ICS 275A - Constraint Networks 33 Examples of generalized arc-consistency x+y+z = 13 implies x<=2, y<=2 Example of relational arc-consistency

Spring 2007 ICS 275A - Constraint Networks 34 More arc-based consistency Global constraints: e.g., all-different constraints Special semantic constraints that appears often in practice and a specialized constraint propagation. Used in constraint programming. Bounds-consistency: pruning the boundaries of domains

Spring 2007 ICS 275A - Constraint Networks 35 Sudoku – Constraint Satisfaction Each row, column and major block must be alldifferent “Well posed” if it has unique solution: 27 constraints Variables: empty slots Domains = {1,2,3,4,5,6,7,8,9} Constraints: 27 all-different Constraint Propagation Inference

Spring 2007 ICS 275A - Constraint Networks 36 Global constraints

Spring 2007 ICS 275A - Constraint Networks 37 Example for alldiff A = {3,4,5,6} B = {3,4} C= {2,3,4,5} D= {2,3,4} E = {3,4} Alldiff (A,B,C,D,E} Arc-consistency does nothing Apply GAC to sol(A,B,C,D,E)?  A = {6}, F = {1}…. Alg: bipartite matching kn^1.5 (Lopez-Ortiz, et. Al, IJCAI-03 pp 245 (A fast and simple algorithm for bounds consistency of alldifferent constraint)

Spring 2007 ICS 275A - Constraint Networks 38 Global constraints Alldifferent Sum constraint (variable equal the sum of others) Global cardinality constraint (a value can be assigned a bounded number of times to a set of variables) The cummulative constraint (related to scheduling tasks)

Spring 2007 ICS 275A - Constraint Networks 39 Bounds consistency

Spring 2007 ICS 275A - Constraint Networks 40 Bounds consistency for Alldifferent constraints

Spring 2007 ICS 275A - Constraint Networks 41 Boolean constraint propagation (A V ~B) and (B) B is arc-consistent relative to A but not vice-versa Arc-consistency by resolution: res((A V ~B),B) = A Given also (B V C), path-consistency: Res((A V ~B),(B V C) = (A V C) What will generalized arc-consistency can do to cnfs? Relational arc-consistency rule = unit-resolution

Spring 2007 ICS 275A - Constraint Networks 42 Boolean constraint propagation If Alex goes, then Becky goes: If Chris goes, then Alex goes: Query: Is it possible that Chris goes to the party but Becky does not? Example: party problem

Spring 2007 ICS 275A - Constraint Networks 43 Gausian and Boolean propagation Linear inequalities Boolean constraint propagation

Spring 2007 ICS 275A - Constraint Networks 44 Constraint propagation for Boolean constraints: Unit propagation

Spring 2007 ICS 275A - Constraint Networks 45 Arc-consistency for non-binary constraints: Generalized arc-consistency (reminder) Complexity: O(t k), t bounds number of tuples. Relational arc-consistency:

Spring 2007 ICS 275A - Constraint Networks 46 Algorithms for relational and generalized arc-cnsistency Think about the following: GAC-i apply AC-i to the dual problem when singleton variables are explicit: the bi-partite representation. What is the complexity? Relational arc-consistency: imitate unit propagation. Apply AC-1 on the dual problemwhere each subset of a scope is presented. Is unit propagation equivalent to AC-4?

Spring 2007 ICS 275A - Constraint Networks 47 Consistency for numeric constraints

Spring 2007 ICS 275A - Constraint Networks 48 Tractable classes

Spring 2007 ICS 275A - Constraint Networks 49 Changes in the network graph as a result of arc-consistency, path-consistency and 4- consistency.

Spring 2007 ICS 275A - Constraint Networks 50 Distributed arc-consistency (Constraint propagation) Implement AC-1 distributedly. Node x_j sends the message to node x_i Node x_i updates its domain: Generalized arc-consistency can be implemented distributedly: sending messages between constraints over the dual graph :

Spring 2007 ICS 275A - Constraint Networks 51 Distributed Arc-Consistency Arc-consistency can be formulated as a distributed algorithm: A BC DF G a Constraint network

Spring 2007 ICS 275A - Constraint Networks 52 A AC AB BCF ABD DFG Relational Arc-consistency A BC DF G The message that R2 sends to R1 is R1 updates its relation and domains and sends messages to neighbors

Spring 2007 ICS 275A - Constraint Networks 53 A AC AB BCF ABD DFG A ABAC ABDBCF DFG B B D F A A A C 1 DRAC on the dual join-graph

Spring 2007 ICS 275A - Constraint Networks 54 Distributed Relational Arc-Consistency DRAC can be applied to the dual problem of any constraint network:

Spring 2007 ICS 275A - Constraint Networks 55 A AC AB BCF ABD DFG A ABAC ABDBCF DFG AB B D B F A A A A C 1 DR-AC on the dual graph A

Spring 2007 ICS 275A - Constraint Networks 56 Node 6 sends messagesNode 5 sends messagesNode 4 sends messagesNode 3 sends messagesNode 2 sends messages A AC AB BCF ABD DFG A A A A 1 3 A B 1 1 A 1 3 B A 1 3 B 1 2 D B 1 1 A 1 3 F D 2 C B 2 C B 1 3 F Iteration A A Node 1 sends messages A ABAC ABDBCF DFG AB B D B F A A A A C A 1 3 A A 1

Spring 2007 ICS 275A - Constraint Networks 57 A AC AB BCF ABD DFG A A A A 1 3 B B A D 1 2 F 1 3 D C 2 B C 2 F 1 3 A B Iteration 1 A ABAC ABDBCF DFG B B D F A A A C 1

Spring 2007 ICS 275A - Constraint Networks 58 A 1 3 AC AB BCF ABD DFG 213 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 1

Spring 2007 ICS 275A - Constraint Networks 59 A 1 3 AC AB BCF ABD DFG 213 A 1 3 A 1 3 A A 1 3 A 1 3 D 2 F 1 3 D 1 2 C 2 B 1 3 C 2 F 1 A B 1 3 B A ABAC ABDBCF DFG B B D F A A A C 1 Iteration B

Spring 2007 ICS 275A - Constraint Networks 60 A 1 3 AC AB BCF 321 ABD DFG 213 Iteration 2 A ABAC ABDBCF DFG B B D F A A A C 1

Spring 2007 ICS 275A - Constraint Networks 61 A 1 3 A 1 3 A 1 3 B 3 A 1 3 D 2 F 1 D 2 C 2 B 1 3 C 2 F 1 A 1 3 A 1 3 A 1 3 AC AB BCF 321 ABD DFG 213 B 1 3 B 1 3 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 3

Spring 2007 ICS 275A - Constraint Networks 62 A 1 3 AC AB 13 BCF 321 ABD DFG 213 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 3

Spring 2007 ICS 275A - Constraint Networks 63 A 1 3 A 1 3 A 1 B 3 A 1 3 D 2 F 1 D 2 C 2 B 3 C 2 F 1 A 1 3 A 1 3 A 1 3 AC AB 13 BCF 321 ABD DFG 213 B 1 3 B 1 3 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 4

Spring 2007 ICS 275A - Constraint Networks 64 A 1 AC AB 13 BCF 321 ABD 132 DFG 213 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 4

Spring 2007 ICS 275A - Constraint Networks 65 A 1 A 1 A 1 B 3 B 3 B 3 A 1 D 2 F 1 D 2 C 2 B 3 C 2 F 1 A 1 A 1 A 1 AC AB 13 BCF 321 ABD 132 DFG 213 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 5

Spring 2007 ICS 275A - Constraint Networks 66 A 1 AC 12 AB 13 BCF 321 ABD 132 DFG 213 A ABAC ABDBCF DFG B B D F A A A C 1 Iteration 5