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 intuitions.]

(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 arc is considered, the domain reduced, and the arcs added to the set to do (similar to the table of Example 4.18 (page 138))
• show explicitly the constraint graph after arc consistency has stopped • show how splitting a domain can be used to solve this problem.