Tipos de Cableados

Let D be a collection of domains and β be any outcome of D. To prove that α dominates D (that is, α _{D β for all β of D), it is sufficient to present evidence that} there is no pos-tree that attests β strictly dominates α, for all β of D. If it is the case, then we have α_<σ β for all σ satisfying Γ, which can be expressed as α D β, for all

β of D.

We look for this evidence in the root node of the pos-tree. In fact, when the root node in a pos-tree σ that satisfies Γ decides α dominates β then we can conclude that σ prefers α over β. If there are no pos-trees that satisfy Γ and also strictly prefer β over α, for any β of D, then we can say that all pos-trees that satisfy Γ prefer α over β, for all β of D. Then, we need to define sufficient conditions that guarantee the root node of every pos-tree that satisfies Γ decides α dominates β, for all β of D.

There are two cases where the root node r reveals that β is not strictly preferred over α, for all β of D:

(i) If α and β have the same value for the variable associated with the root node (i.e., Yr), then r should have no children associated with the assignment Yr =

α(Yr). This is guaranteed by having α(Yr) and y are AY>-equivalent for some

y ∈ Yr− {α(Y )} (see Section 3.4.2.1 of Chapter 3);

(ii) if α(Yr) 6= β(Yr), then α(Yr) should be better than the possible values that β

can have for Yr(i.e., α(Yr) <ry, for all y ∈ D(Yr) − {α(Y )}).

Therefore, we say that α root-dominates a collection of domains D if: for all Y /∈S

5.3. Dominance Pruning Rules

(i) α(Y )_AY

> y for all y ∈ D(Y ) − {α(Y )};

(ii) if α(Y ) ∈ D(Y ) then α(Y ) and y are_AY_>-equivalent for some y ∈ Y −{α(Y )}. The following result states the soundness of the root-dominates rule.

Proposition 2. If_{α root-dominates D then α dominates D, i.e., α D β for all β of D.} Proof: _{Let σ be any pos-tree satisfying Γ and let β be any element of D. Let r be} the root node of σ. Let us look into σ and focus more particularly on the root node r that decides α and β. Node r has an associated variable Yr. Yrcannot belong to Wϕ

of any preference statement ϕ satisfying Γ and written as uϕ : xϕ > x0ϕ[Wϕ]. This is

because there will be at least one variable (Xϕ) that should appear before variables

in Wϕin σ (Proposition 1(2) in Section 5.3). There are two possibilities:

• α(Yr) 6= β(Yr): Then r decides α and β. Condition (i) implies α(Y ) AY> β(Y ).

This implies α(Yr) and β(Yr). Thus, α and β differ on the root node r with α

having better value on Yr. Then, we can conclude α<σ β.

• α(Yr) = β(Yr): This implies α(Yr) ∈ D(Yr) (since β(Yr) ∈ D(Yr) as β is

of D), and so condition (ii) implies that α(Y ) is _{>-equivalent to some other} element of Y . A root node with such equivalence cannot have children in σ (by Definition of a cp-tree - see Definition 11 in Section 3.4.2.1). Therefore, β is not strictly preferred over α by σ. Thus, since node r decides α and β so again α(Yr) <r β(Yr).

Then, we have shown that α _{D β for all pos-trees σ that satisfy Γ. Thus, we can} conclude that α dominates D as β was an arbitrary element of D.



Example 7. LetV be the set of variables {X, Y, Z} with (initial) domains defined as follows: X = {x1, x2, x3, x4}, Y = {y1, y2} and Z = {z1, z2}. Let cp-theory Γ consist

of the five statements> : x1 > x3,> : x2 > x3, and> : x2 > x4 [{Z}] x1 : y1 > y2,

andx2 : y2 > y1.

Suppose a CSP search tree is created and used to find optimal solutions (with respect to Γ). Suppose the variables are instantiated in the following order: X, Y then Z. This instantiation order is compatible with the user preferences, as should be the values’ instantiation order. Let us assume that at some node of this search tree, we have a previously found optimal assignment α = x2y2z2 and the variables’ domains are as

follows. LetD(X) = {x3}, D(Y ) = {y2} and D(Z) = {z1} with a partial assignment

represented by a tupleb = x3y2z1. We want to check whetherα dominates the collection

we can avoid expanding the search tree and backtrack. We first look for all variables not inS

ϕ∈ΓWϕ. We can see that

S

ϕ∈ΓWϕ = {Z}. Now, we need to find out whether

the sufficient conditions for α to root-dominate D are verified. For variable X, the preference statements in _{Γ show A}X_>= {(x1, x3), (x2, x3), (x2, x4)} as only the three

first statements concern the preference over the values of _{X, A}Y_>= {(y1, y2), (y2, y1)}

as only the fourth and fifth statements concern the preference over the values of _{Y .} Thus, we can see that _{α(X) = x2 A}X_{> x3}. For variable_{Y , α(Y ) = y2 ∈ D(Y ), and} y2 AY> y1 AY> y2, soα(Y ) and y1 areAY>-equivalent. Hence,α root-dominates D.

Then, we backtrack and the collection of domains will be as follows. D(X) = {x3},

D(Y ) = {y2} and D(Z) = {z1, z2} with b = x3y2. We can notice that only D(Z)

changes butZ 6∈S

ϕ∈ΓWϕ. So,α still root-dominates D.

The search backtracks again so that we haveD(X) = {x3}, D(Y ) = {y1, y2} and

D(Z) = {z1, z2} with b = x3. OnlyD(Y ) changes. The fourth and fifth statements in Γ

showα(Y ) = y2 AY>y1 A>Y y2, which leads to confirm thatα(Y ) = y2(y2 ∈ D(Y )) is

AY>-equivalent with another valuey1 ∈ Y − {α(Y )}. Thus, α root-dominates D again.