6. Consider a scheduling problem, where there are five activities to be scheduled in four time slots. Suppose we represent the activities by the variables A, B, C, D, and E, where the domain of each variable is {1, 2, 3, 4} and the constraints are A > D, D > E, C ≠ A, C > E, C ≠ D, B ≥ A, B ≠ C, and C ≠ D + 1.

[Before you start this, try to find the legal schedule(s) using your own intutions.]

(a) Show how backtracking solves this problem. To do this, you should draw the search tree generated to find all answers. Indicate clearly the valid schedule(s). Make sure you choose a reasonable variable ordering.

To indicate the search tree, write it in text form with each branch on one line. For example, suppose we had variables X, Y, and Z with domains t, f and constraints X ≠ Y and Y ≠ Z. The corresponding search tree is written as:

X=t Y=t failure Y=f Z=t solution Z=f failure X=f Y=t Z=t failure Z=f solution Y=f failure

[Hint: It may be easier to write a program to generate such a tree for a particular problem than to do it by hand.]

(b) Show how arc consistency solves this problem. To do this you must

• draw the constraint graph;

• show which elements of a domain are deleted at each step, and which arc is responsible for removing the element;

• show explicitly the constraint graph after arc consistency has stopped; and

• show how splitting a domain can be used to sove this problem.