Directional consistency Chapter 4

1. Directional consistencyChapter 4 ICS-275 Spring 2009 ICS 275 - Constraint Networks

2. Backtrack-free search: orWhat level of consistency will guarantee global-consistency ICS 275 - Constraint Networks

3. Directional arc-consistency:another restriction on propagation D4={white,blue,black} D3={red,white,blue} D2={green,white,black} D1={red,white,black} X1=x2, x1=x3,x3=x4 ICS 275 - Constraint Networks

4. Directional arc-consistency:another restriction on propagation D4={white,blue,black} D3={red,white,blue} D2={green,white,black} D1={red,white,black} X1=x2, x1=x3,x3=x4 ICS 275 - Constraint Networks

5. Directional arc-consistency:another restriction on propagation • D4={white,blue,black} • D3={red,white,blue} • D2={green,white,black} • D1={red,white,black} • X1=x2, x1=x3, x3=x4 • After DAC: • D1= {white}, • D2={green,white,black}, • D3={white,blue}, • D4={white,blue,black} ICS 275 - Constraint Networks

6. Algorithm for directional arc-consistency (DAC) • Complexity: ICS 275 - Constraint Networks

7. Directional arc-consistency may not be enough Directional path-consistency ICS 275 - Constraint Networks

8. Algorithm directional path consistency (DPC) ICS 275 - Constraint Networks

9. Example of DPC ICS 275 - Constraint Networks

10. Directional i-consistency ICS 275 - Constraint Networks

11. Algorithm directional i-consistency ICS 275 - Constraint Networks

12. Width along ordering d, w(d): max # of previous parents Induced width w*(d): The width in the ordered induced graph Induced-width w*: Smallest induced-width over all orderings Finding w* NP-complete(Arnborg, 1985) but greedy heuristics (min-fill). The induced-widthDPC recursively connects parents in the ordered graph, yielding: ICS 275 - Constraint Networks

13. Induced-width ICS 275 - Constraint Networks

14. Examples Three orderings : d1 = (F,E,D,C,B,A), its reversed ordering d2 = (A,B,C,D,E, F), and d3 = (F,D,C,B,A,E). Orderings from bottom to top, The parents of A along d1 are {B,C,E}. Width of A along d1 is 3, of C along d1 is 1, of A along d3 is 2. w(d1) = 3, w(d2) = 2, and w(d3) = 2. The width of graph G is 2. ICS 275 - Constraint Networks

15. Induced-width and DPC • The induced graph of (G,d) is denoted (G*,d) • The induced graph (G*,d) contains the graph generated by DPC along d, and the graph generated by directional i-consistency along d ICS 275 - Constraint Networks

16. Refined Complexity using induced-width • Consequently we wish to have ordering with minimal induced-width • Induced-width is equal to tree-width to be defined later. • Finding min induced-width ordering is NP-complete ICS 275 - Constraint Networks

17. Greedy algorithms for iduced-width • Min-width ordering • Max-cardinality ordering • Min-fill ordering • Chordal graphs ICS 275 - Constraint Networks

18. Min-width ordering ICS 275 - Constraint Networks

19. Min-induced-width ICS 275 - Constraint Networks

20. Min-fill algorithm • Prefers a node who add the least number of fill-in arcs. • Empirically, fill-in is the best among the greedy algorithms (MW,MIW,MF,MC) ICS 275 - Constraint Networks

21. Cordal graphs and Max-cardinality ordering • A graph is cordal if every cycle of length at least 4 has a chord • Finding w* over chordal graph is easy using the max-cardinality ordering • If G* is an induced graph it is chordal • K-trees are special chordal graphs. • Finding the max-clique in chordal graphs is easy (just enumerate all cliques in a max-cardinality ordering ICS 275 - Constraint Networks

22. Example We see again that G in Figure 4.1(a) is not chordal since the parents of A are not connected in the max-cardinality ordering in Figure 4.1(d). If we connect B and C, the resulting induced graph is chordal. ICS 275 - Constraint Networks

23. Max-caedinality ordering Figure 4.5 The max-cardinality (MC) ordering procedure. ICS 275 - Constraint Networks

24. Width vs local consistency:solving trees ICS 275 - Constraint Networks

25. Tree-solving ICS 275 - Constraint Networks

26. Width-2 and DPC ICS 275 - Constraint Networks

27. Width vs directional consistency(Freuder 82) ICS 275 - Constraint Networks

28. Width vs i-consistency • DAC and width-1 • DPC and width-2 • DIC_i and with-(i-1) •  backtrack-free representation • If a problem has width 2, will DPC make it backtrack-free? • Adaptive-consistency: applies i-consistency when i is adapted to the number of parents ICS 275 - Constraint Networks

29. Adaptive-consistency ICS 275 - Constraint Networks

30. Bucket EliminationAdaptive Consistency (Dechter & Pearl, 1987) = ¹ D = C A ¹ C B = A contradiction = Bucket E: E ¹ D, E ¹ C Bucket D: D ¹ A Bucket C: C ¹ B Bucket B: B ¹ A Bucket A: ICS 275 - Constraint Networks

31. E D || RDCB C || RACB || RAB B RA A A || RDB D C B E Bucket EliminationAdaptive Consistency (Dechter & Pearl, 1987) RCBE || RDBE , || RE ICS 275 - Constraint Networks

32. eliminating E C RDBC D 3 value assignment B The Idea of Elimination ICS 275 - Constraint Networks

33. Adaptive-consistency, bucket-elimination ICS 275 - Constraint Networks

34. Properties of bucket-elimination(adaptive consistency) • Adaptive consistency generates a constraint network that is backtrack-free (can be solved without dead-ends). • The time and space complexity of adaptive consistency along ordering d is respectively, or O(r k^(w*+1)) when r is the number of constraints. • Therefore, problems having bounded induced width are tractable (solved in polynomial time) • Special cases: trees ( w*=1 ), series-parallel networks (w*=2 ), and in general k-trees ( w*=k ). ICS 275 - Constraint Networks

35. Back to Induced width • Finding minimum-w* ordering is NP-complete (Arnborg, 1985) • Greedy ordering heuristics: min-width, min-degree, max-cardinality (Bertele and Briochi, 1972; Freuder 1982), Min-fill. ICS 275 - Constraint Networks

36. Solving Trees (Mackworth and Freuder, 1985) Adaptive consistency is linear for trees and equivalent to enforcing directionalarc-consistency (recording only unary constraints) ICS 275 - Constraint Networks

37. E E E E D C D D D C C C B B B B A Summary: directional i-consistency Adaptive d-path d-arc ICS 275 - Constraint Networks

38. Variable Elimination Eliminate variables one by one: “constraint propagation” Solution generation after elimination is backtrack-free ICS 275 - Constraint Networks

39. Relational consistency(Chapter 8) • Relational arc-consistency • Relational path-consistency • Relational m-consistency • Relational consistency for Boolean and linear constraints: • Unit-resolution is relational-arc-consistency • Pair-wise resolution is relational path-consistency ICS 275 - Constraint Networks

40. Sudoku’s propagation • http://www.websudoku.com/ • What kind of propagation we do? ICS 275 - Constraint Networks

41. Complexity of Adaptive-consistency as a function of the hypergraph • Processing a bucket is also exponential in its number of constraints. • The number of constraints in a bucket is bounded by the total, r, number of functions. Can we have a better bound? • Theorem: If we combine buckets into superbuckets so variables in a superbucket are covered by original constraints scopes, then “super-bucket elimination is exp in the max number of constraints in a superbucket. • Hyper-induced-width: The number of constraint in a super-bucket ICS 275 - Constraint Networks