Directional consistency Chapter 4 ICS275 Spring 2009 ICS

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

Backtrack-free search: or What level of consistency will guarantee globalconsistency Spring 2009 ICS 275 - Constraint Networks 2

Directional arc-consistency: another restriction on propagation D 4={white, blue, black} D 3={red, white, blue} D 2={green, white, black} D 1={red, white, black} X 1=x 2, x 1=x 3, x 3=x 4 Spring 2009 ICS 275 - Constraint Networks 3

Directional arc-consistency: another restriction on propagation D 4={white, blue, black} D 3={red, white, blue} D 2={green, white, black} D 1={red, white, black} X 1=x 2, x 1=x 3, x 3=x 4 Spring 2009 ICS 275 - Constraint Networks 4

Directional arc-consistency: another restriction on propagation l l l l l D 4={white, blue, black} D 3={red, white, blue} D 2={green, white, black} D 1={red, white, black} X 1=x 2, x 1=x 3, x 3=x 4 After DAC: D 1= {white}, D 2={green, white, black}, D 3={white, blue}, D 4={white, blue, black} Spring 2009 ICS 275 - Constraint Networks 5

Algorithm for directional arcconsistency (DAC) l Complexity: Spring 2009 ICS 275 - Constraint Networks 6

Directional arc-consistency may not be enough Directional path-consistency Spring 2009 ICS 275 - Constraint Networks 7

Algorithm directional path consistency (DPC) Spring 2009 ICS 275 - Constraint Networks 9

Example of DPC Spring 2009 ICS 275 - Constraint Networks 10

Directional i-consistency Spring 2009 ICS 275 - Constraint Networks 11

Algorithm directional i-consistency Spring 2009 ICS 275 - Constraint Networks 13

The induced-width DPC recursively connects parents in the ordered graph, yielding: l l Width along ordering d, w(d): • Induced width w*(d): • The width in the ordered induced graph Induced-width w*: • Smallest induced-width over all orderings Finding w* • Spring 2009 max # of previous parents NP-complete (Arnborg, 1985) but greedy heuristics (min-fill). ICS 275 - Constraint Networks 14

Induced-width Spring 2009 ICS 275 - Constraint Networks 15

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 275 - Constraint Networks 16

Induced-width and DPC l l 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 iconsistency along d Spring 2009 ICS 275 - Constraint Networks 17

Refined Complexity using induced-width l l l 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 275 - Constraint Networks 18

Greedy algorithms for iducedwidth • Min-width ordering • Max-cardinality ordering • Min-fill ordering • Chordal graphs Spring 2009 ICS 275 - Constraint Networks 19

Min-width ordering Spring 2009 ICS 275 - Constraint Networks 20

Min-induced-width Spring 2009 ICS 275 - Constraint Networks 21

Min-fill algorithm l l 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 275 - Constraint Networks 22

Cordal graphs and Maxcardinality ordering l l l 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 maxcardinality ordering Spring 2009 ICS 275 - Constraint Networks 23

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 275 - Constraint Networks 24

Max-caedinality ordering Figure 4. 5 The max-cardinality (MC) ordering procedure. Spring 2009 ICS 275 - Constraint Networks 25

Width vs local consistency: solving trees Spring 2009 ICS 275 - Constraint Networks 26

Tree-solving Spring 2009 ICS 275 - Constraint Networks 27

Width-2 and DPC Spring 2009 ICS 275 - Constraint Networks 28

Width vs directional consistency (Freuder 82) Spring 2009 ICS 275 - Constraint Networks 29

Width vs i-consistency l l l 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 275 - Constraint Networks 30

Adaptive-consistency Spring 2009 ICS 275 - Constraint Networks 31

Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) = ¹ = 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: Spring 2009 D=C A¹C B=A contradiction ICS 275 - Constraint Networks 32

Bucket Elimination Adaptive Consistency (Dechter & Pearl, 1987) E D || RDCB || RAB C B RA A A || RDB D || RDBE , RCBE C || RE B E Spring 2009 ICS 275 - Constraint Networks 33

The Idea of Elimination eliminating E C D RDBC 3 value assignment B Spring 2009 ICS 275 - Constraint Networks 34

Adaptive-consistency, bucket-elimination Spring 2009 ICS 275 - Constraint Networks 35

Properties of bucket-elimination (adaptive consistency) l Adaptive consistency generates a constraint network that is backtrack-free (can be solved without dead-ends). l 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. l Therefore, problems having bounded induced width are tractable (solved in polynomial time) l Special cases: trees ( w*=1 ), series-parallel networks (w*=2 ), and in general k-trees ( w*=k ). Spring 2009 ICS 275 - Constraint Networks 36

Back to Induced width l l 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 275 - Constraint Networks 37

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 275 - Constraint Networks 38

Summary: directional i-consistency E D E C B A D C D B Adaptive Spring 2009 E C B d-path ICS 275 - Constraint Networks E D C B d-arc 39

Variable Elimination Eliminate variables one by one: “constraint propagation” Solution generation after elimination is backtrack-free Spring 2009 ICS 275 - Constraint Networks 40

Relational consistency (Chapter 8) l l l Relational arc-consistency Relational path-consistency Relational m-consistency l Relational consistency for Boolean and linear constraints: • Unit-resolution is relational-arc-consistency • Pair-wise resolution is relational pathconsistency Spring 2009 ICS 275 - Constraint Networks 41

Sudoku’s propagation l l http: //www. websudoku. com/ What kind of propagation we do? Spring 2009 ICS 275 - Constraint Networks 42

Complexity of Adaptive-consistency as a function of the hypergraph l l 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 Spring 2009 ICS 275 - Constraint Networks 43