1 Directional consistency Chapter 4 ICS-275 Spring 2009 ICS Constraint Networks

Spring 2009 ICS Constraint Networks 2 Backtrack-free search: or What level of consistency will guarantee global- consistency

Spring 2009 ICS 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

Spring 2009 ICS 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

Spring 2009 ICS 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}

Spring 2009 ICS Constraint Networks 6 Algorithm for directional arc- consistency (DAC) Complexity :

Spring 2009 ICS Constraint Networks 7 Directional arc-consistency may not be enough  Directional path-consistency

Spring 2009 ICS Constraint Networks 8 Directional path-consistency

Spring 2009 ICS Constraint Networks 9 Algorithm directional path consistency (DPC)

Spring 2009 ICS Constraint Networks 10 Example of DPC

Spring 2009 ICS Constraint Networks 11 Directional i-consistency

Spring 2009 ICS Constraint Networks 12 Example

Spring 2009 ICS Constraint Networks 13 Algorithm directional i- consistency

Spring 2009 ICS Constraint Networks 14 The induced-width DPC recursively connects parents in the ordered graph, yielding: 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).

Spring 2009 ICS Constraint Networks 15 Induced-width

Spring 2009 ICS Constraint Networks 16 Examples Three orderings : d 1 = (F,E,D,C,B,A), its reversed ordering d 2 = (A,B,C,D,E, F), and d 3 = (F,D,C,B,A,E). Orderings from bottom to top, The parents of A along d 1 are {B,C,E}. Width of A along d 1 is 3, of C along d 1 is 1, of A along d 3 is 2. w(d 1 ) = 3, w(d 2 ) = 2, and w(d 3 ) = 2. The width of graph G is 2.

Spring 2009 ICS Constraint Networks 17 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

Spring 2009 ICS Constraint Networks 18 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

Spring 2009 ICS Constraint Networks 19 Greedy algorithms for iduced- width Min-width ordering Max-cardinality ordering Min-fill ordering Chordal graphs

Spring 2009 ICS Constraint Networks 20 Min-width ordering

Spring 2009 ICS Constraint Networks 21 Min-induced-width

Spring 2009 ICS Constraint Networks 22 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)

Spring 2009 ICS Constraint Networks 23 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

Spring 2009 ICS Constraint Networks 24 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.

Spring 2009 ICS Constraint Networks 25 Max-caedinality ordering Figure 4.5 The max-cardinality (MC) ordering procedure.

Spring 2009 ICS Constraint Networks 26 Width vs local consistency: solving trees

Spring 2009 ICS Constraint Networks 27 Tree-solving

Spring 2009 ICS Constraint Networks 28 Width-2 and DPC

Spring 2009 ICS Constraint Networks 29 Width vs directional consistency (Freuder 82)

Spring 2009 ICS Constraint Networks 30 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

Spring 2009 ICS Constraint Networks 31 Adaptive-consistency

Spring 2009 ICS Constraint Networks 32 Bucket E: E  D, E  C Bucket D: D  A Bucket C: C  B Bucket B: B  A Bucket A: A  C contradiction = D = C B = A Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) = 

Spring 2009 ICS Constraint Networks 33 A E D C B E A D C B || R D BE, || R E || R DB || R DCB || R ACB || R AB RARA R C BE Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987)

Spring 2009 ICS Constraint Networks 34 The Idea of Elimination 3 value assignment D B C R DBC eliminating E

Spring 2009 ICS Constraint Networks 35 Adaptive- consistency, bucket-elimination

Spring 2009 ICS Constraint Networks 36 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 ).

Spring 2009 ICS Constraint Networks 37 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.

Spring 2009 ICS Constraint Networks 38 Solving Trees (Mackworth and Freuder, 1985) Adaptive consistency is linear for trees and equivalent to enforcing directional arc-consistency (recording only unary constraints)

Spring 2009 ICS Constraint Networks 39 Summary: directional i-consistency A E CD B D C B E D C B E D C B E Adaptived-arcd-path

Spring 2009 ICS Constraint Networks 40 Variable Elimination Eliminate variables one by one: “constraint propagation” Solution generation after elimination is backtrack-free

Spring 2009 ICS Constraint Networks 41 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

Spring 2009 ICS Constraint Networks 42 Sudoku’s propagation What kind of propagation we do?

Spring 2009 ICS Constraint Networks 43 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