1. Directional consistency Chapter 4
Roberto Posenato
November 2018

2. background
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3. Ordered graphs and Induced-Width
d1= F,E,D,C,B,A
d2= A,B,C,D,E,F
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4. Ordered graphs and Induced-Width
Ordering d1= F,E,D,C,B,A
d2= A,B,C,D,E,F
d3= F,D,C,B,A,E
Ordered graph w.r.t. a d = node in line from the last (top) to the first (bottom)
Parents of v = {adjacent nodes preceding v in the ordering}
Width of a node in an order = #Parents in an ordering
Width of an ordering d = w(d) = max width over all nodes
Width of a graph = min width over all orderings
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5. Ordered graphs and Induced-Width
Induced Graph (G*,d) of an ordered one (G,d) From top to bottom, all parents of a node are connected. (dashed edges)
Induced width w*(d) The width in the ordered induced graph
Induced-width w* min induced-width over all orderings
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6. Ordered graphs and Induced-Width
Interesting properties
A graph is a tree iff has induced width of 1
A greedy algorithm determines the min width of a graph in time O(n2)
Finding the minimum induced width of a graph is NP-hard (decision version is NP-complete (Arnborg, 1985))
Check the book for two greedy algorithms!
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7. Backtrack-free search
Previously, we have seen that a globally consistent network guarantees backtrack-free search
Globally consistency is based bounding the number of variables that participate in an inference
An orthogonal approach is to restrict inference relative to a given ordering of the variables (used also for the search of solutions)
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8. 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
We want to determine solutions considering order (x1, x2, x3, x4)
After DAC:
D1= {white},
D2={green, white, black},
D3={white, blue},
D4={white, blue, black}
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9. Directional arc-consistency: another restriction on propagation
D1= {white},
D2= {green, white, black},
D3= {white, blue},
D4= {white, blue, black}
is (x1, x2, x3, x4) directional arc-consistent
but it is not arc-consistent!
(Check R31!)
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10. Algorithm for directional arc-consistency (DAC)
<= From last variable!
Main advantage: each constraint is processed exactly one time!
Complexity: O(e k2) (same as full arc-consistency)
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11. Directional arc-consistency may not be enough  Directional path-consistency
4 variables, common domain D = {red, blue}
Rij = {(ai, aj) | ai ≠ aj }
We want to determine solutions considering order d = (x1, x2, x3, x4)
The network is already full arc-consistent
By definition, it is also d-directional arc-consistent
It is sufficient to guarantee a backtrack-free search for a consistent solution?
NO! Try to find an assignment for x3!
Higher levels of consistency may be necessary!
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12. Algorithm directional path consistency (DPC)
The complexity is equal to the full path consistency problem
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13. Directional path-consistency for graph-coloring
4 variables, common domain D = {red, blue}
Rij = {(ai, aj) | ai ≠ aj } limited to R1,2, R2,3,, R3,4, and R1,4
We want to determine solutions considering order d = (x1, x2, x3, x4)
Enforcing full path-consistency requires adding constraint R1,3 = {x1 = x3} and R2,4 = {x2 = x4}
Directional path-consistency relative to d requires constraint R1,3 = {x1 = x3}
This allow a solution to be assembled along d without encountering dead-ends.
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14. Example of DPC
Try it yourself:
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15. Directional i-consistency
Till now directional consistency was defined considering binary network
Directional i–consistency must account for constraints with larger scopes
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16. Algorithm directional i-consistency
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17. Graphs Aspects of Directional Consistency
During DPC, a variable x only affects the constraints between a pair of earlier variables
A new constraint (and a new arc) may be added to the graph
DPC recursively connects the parents of every two nodes in the ordered constraint network
The induced graph (G*,d) contains the graph generated by DPC along d, and the graph generated by directional i-consistency along d
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18. Refined Complexity using induced-width
Consequently we wish to have ordering with minimal induced-width… but
Finding min induced-width ordering is NP-complete
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19. Greedy algorithms for induced-width
Min-induced-width ordering
Max-cardinality ordering
Min-fill ordering
Chordal graphs
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20. Min-induced-width
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21. Width vs local consistency
We have seen examples where DPC changes a network so that a solution can be found in a backtrack-free manner…
but there are also many examples in which even the full path-consistency is not sufficient for finding solutions in a backtrack-free manner
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22. Width vs local consistency
It would be nice to have a criterion that could say, in advance, the level of consistency sufficient for finding solutions in a backtrack-free manner!
Well, such criterion exists and it is based on induced width of the network graph
Here we consider only two simple cases
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23. Width vs local consistency: induced width 1
A constraint graph has width 1 iff is a tree
Let d an ordering having width 1.
This means that the value of a variable may depend on the value of only one previous variable in d.
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24. Tree-solving algorithm
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25. Width vs local consistency: induced width 2
It is possible to find a solution in a backtrack free way also for graph having width 2
It is necessary to use both DAC and DPC
Let d an ordering having width 2.
This means that the value of a variable may depend on the values of two previous variables in d.
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26. Width vs local consistency: induced width 2
BE CAREFUL!
If a graph has width 2 BUT it is not directional path-consistent, then DPC may make it adding arcs!
Adding arcs may increase the width!
So, for having a backtrack free searching it is necessary that the induced width must be 2!
Theorem 4.6
A binary constraint network having an induced width of 2 can be solved in O(n k3)
Modifying algorithm MIN-INDUCED-WIDTH, it is possible to say if a graph has induced width of 2 in linear time!
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27. Width vs directional consistency (Freuder 82)
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28. Width vs directional consistency
DAC and width-1
DPC and width-2
DIC_i and with-(i-1)
 backtrack-free representation
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