Chapter 4 ICS-275 Fall 2010 Fall 2010 ICS 275 - Constraint - PowerPoint PPT Presentation

Directional consistency Chapter 4 ICS-275 Fall 2010 Fall 2010 ICS 275 - Constraint Networks 1

Tractable Tractable classes classes Fall 2010 2

Backtrack-free search: or What level of consistency will guarantee global- consistency Backtrack free and queries: Consistency, All solutions Counting optimization Fall 2010 3 ICS 275 - Constraint Networks

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 Fall 2010 4 ICS 275 - Constraint Networks

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}  Fall 2010 5 ICS 275 - Constraint Networks

Algorithm for directional arc- consistency (DAC)  Complexity : 2 O ( ek ) Fall 2010 6 ICS 275 - Constraint Networks

Directional arc-consistency may not be enough  Directional path-consistency Fall 2010 7 ICS 275 - Constraint Networks

Algorithm directional path consistency (DPC) Fall 2010 8 ICS 275 - Constraint Networks

Example of DPC { 1 , 2 }   E { 1 , 2 , 3 } { 1 , 2 }  D C    B A { 1 , 2 } { 1 , 2 } Fall 2010 9 ICS 275 - Constraint Networks

Directional i-consistency Fall 2010 10 ICS 275 - Constraint Networks

Algorithm directional i- consistency Fall 2010 11 ICS 275 - Constraint Networks

The induced-width DPC recursively connects parents in the ordered graph, yielding: Width along ordering d , w(d):  E • max # of previous parents D C Induced width w*(d):  • The width in the ordered A B induced graph Induced-width w*:  • Smallest induced-width over all orderings Finding w*  • NP-complete (Arnborg, 1985) but greedy heuristics (min-fill). Fall 2010 12 ICS 275 - Constraint Networks

Induced-width Fall 2010 13 ICS 275 - Constraint Networks

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. Fall 2010 14 ICS 275 - Constraint Networks

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 Fall 2010 15 ICS 275 - Constraint Networks

Greedy algorithms for induced-width • Min-width ordering • Max-cardinality ordering • Min-fill ordering • Chordal graphs Fall 2010 16 ICS 275 - Constraint Networks

Min-width ordering Fall 2010 17 ICS 275 - Constraint Networks

Min-induced-width Fall 2010 18 ICS 275 - Constraint Networks

Min-fill algorithm  Prefers a node who adds the least number of fill-in arcs.  Empirically, fill-in is the best among the greedy algorithms (MW,MIW,MF,MC) Fall 2010 19 ICS 275 - Constraint Networks

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 Fall 2010 20 ICS 275 - Constraint Networks

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. Fall 2010 21 ICS 275 - Constraint Networks

Max-cardinality ordering Figure 4.5 The max-cardinality (MC) ordering procedure. Fall 2010 22 ICS 275 - Constraint Networks

Width vs local consistency: solving trees Fall 2010 23 ICS 275 - Constraint Networks

Tree-solving 2 complexity : O ( nk ) Fall 2010 24 ICS 275 - Constraint Networks

Width-2 and DPC Fall 2010 25 ICS 275 - Constraint Networks

Width vs directional consistency (Freuder 82) Fall 2010 26 ICS 275 - Constraint Networks

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 Fall 2010 27 ICS 275 - Constraint Networks

Adaptive-consistency Fall 2010 28 ICS 275 - Constraint Networks

Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) =  = Bucket E: E  D, E  C Bucket D: D  A D = C Bucket C: C  B A  C Bucket B: B  A B = A Bucket A: contradiction * O(n exp(w )) Complexity : * w - induced width Fall 2010 29 ICS 275 - Constraint Networks

Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) E    Bucket ( E ) : E D, E C, E B D  { 1 , 2 } || R DCB Bucket ( D ) : D A   E  C || R ACB Bucket ( C ) : C B { 1 , 2 , 3 } { 1 , 2 }  || R AB  Bucket ( B ) : B A B D C   R A Bucket ( A ) : A    B A Bucket ( A ) : A D, A B A { 1 , 2 } { 1 , 2 }  || R DB Bucket ( D ) : D E D   Bucket ( C ) : C B , C E || R D R C  BE , C Bucket ( B ) : B E BE || R E Bucket ( E ) : B E * O(n exp(w (d))) Time and space : , * w (d) - induced width along ordering d Fall 2010 30 ICS 275 - Constraint Networks

The Idea of Elimination eliminating E C R DBC D 3 value assignment B   R R R R DBC ED EB EC DBC  Eliminate variable E join and project Fall 2010 31 ICS 275 - Constraint Networks

Variable Elimination Eliminate variables one by one: “constraint propagation” Solution generation 3 after elimination is backtrack-free Fall 2010 32 ICS 275 - Constraint Networks

Adaptive- consistency , bucket-elimination Fall 2010 33 ICS 275 - Constraint Networks

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    w * 1 w * 1 along ordering d is respectively, O (n (2 k) ), O (n (k) 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 ). Fall 2010 34 ICS 275 - Constraint Networks

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. Fall 2010 35 ICS 275 - Constraint Networks

Solving Trees (Mackworth and Freuder, 1985) Adaptive consistency is linear for trees and equivalent to enforcing directional arc-consistency (recording only unary constraints) Fall 2010 36 ICS 275 - Constraint Networks

Summary: directional i-consistency E E  E E  D C  D D D     C C C B B B A B Adaptive d-path d-arc    E : E D, E C, E B R R D C R , R   D : D C, D A DCB D B D R R  C : C B CB C R  B : A B D A : Fall 2010 37 ICS 275 - Constraint Networks

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 Fall 2010 38 ICS 275 - Constraint Networks

Sudoku’s propagation  http://www.websudoku.com/  What kind of propagation we do? Fall 2010 39 ICS 275 - Constraint Networks

Sudoku • Variables: 81 slots • Domains = {1,2,3,4,5,6,7,8,9} • Constraints: • 27 not-equal Constraint propagation 2 3 2 4 6 Each row, column and major block must be alldifferent “Well posed” if it has unique solution: 27 constraints

Sudoku Each row, column and major block must be alldifferent “Well posed” if it has unique solution