Assume an Element constraint for \(T[y]=z\), with:

• \(D(y) = \{0,1,2,3,4\}\)
• \(T = [5,5,5,10,8]\)
• \(D(z) = \{5,10\}\)

What is the domain of \(y\) after a domain-consistent filtering?

Assume an Element constraint for \(T[x][y]=z\), with:

• \(D(x) = \{0,1,2\}\)
• \(D(y) = \{3\}\)
• \(T = [[1,8,9,6],[1,9,2,4],[9,8,9,8],[1,9,2,5]]\)
• \(D(z) = \{2,...,6\}\)

What is the domain of \(z\) after a domain-consistent filtering?

Assume an Element constraint for \(T[y]=z\), with:

• \(D(y) = \{0,1,2,3,4\}\)
• \(T = [5,5,5,10,8]\)
• \(D(z) = \{5,6,7,8,9,10\}\)

What is the domain of \(z\) after a domain-consistent filtering?

In the stable matching problem, the constraint "the company of the student of company c is c" is modeled with:

We consider the \(\mathit{AllDifferent}(x_1,x_2,x_3,x_4)\) constraint with domains \(D(x_1)=\{1,3,4\}, D(x_2)=\{1,3\}, D(x_3)=\{1,3\}, D(x_4)=\{1,4,5\}\). This constraint is satisfied if and only if all the values taken by \(x_1,x_2,x_3,x_4\) are different. Select all the correct statements:

A binary constraint \(C(X,Y)\) is represented visually (feasible solutions inside the green polygons).

The domains \(D(X)\) and \(D(Y)\) are also represented as small green rectangles on the axes (the lower and upper bounds are extended in dashed red).

The image displays the domains AFTER they have been pruned by the constraint.

Select all the true statements given the current represented domains:

Select all the correct statements given a constraint \(C(X,Y,Z)\) with three finite-domain integer variables in its scope:

Bound and Domain consistencies are not the only form of consistency. We define next the range-consistency. A constraint \(C\) is range-consistent if, and only if:

\(\forall 1 \leq i \leq n, \forall v_i \in D(i):\quad \exists (v_1,\dots,v_{i-1},v_{i+1},\dots,v_n) \in\) \([\min(D_1),\max(D_1)] \times \cdots \times [\min(D_{i-1}),\max(D_{i-1})] \times [\min(D_{i+1}),\max(D_{i+1})] \times \cdots \times [\min(D_n),\max(D_n)]\) \(\text{ such that } C(v_1,\dots,v_i,\ldots,v_n)\)

Select all the correct statements:

Consider the constraint \(X+Y=0\) with \(D(X)=[100,110]\) and \(D(Y)=[-1000,120] \cap [-105, 1000]\). Assuming a consistent (does not remove any solution) filtering algorithm for the constraint that achieves bound consistency (after its filtering the constraint is bound consistent), what will be the domain, \(D(X)\), after filtering?

Write your answer in the format \(\{v_1,v_2,...,v_n\}\) with \(v_i < v_i+1\). For instance, write \(\{6,10,13\}\) but not \(\{13,6,10\}\).