Question

------- 6.2 is not the question -------

6.2 Consider the problem of placing k knights on an n X n chessboard such that no two knights are attacking each other, where k is given and k ≤ n^2

a. Choose a CSP formulation. In your formulation, what are the variables?

b. What are the possible values of each variable?

c. What sets of variables are constrained, and how?

d. Now consider the problem of putting as many knights as possible on the board without any attacks. Explain how to solve this with local search by defining appropriate

ACTIONS and RESULT functions and a sensible objective function.

-----------------------------

3. Consider the CSP from exercise 6.2 and your formulation (before part d). Let n = 3 and k = 4. If your formulation is not node consistent make it node consistent. Perform the AC-3 algorithm by hand to ensure your formulation is arch consistent.

4. Perform Backtracking search, by hand, on the resulting CSP from question 3.

Answer 1

6.2

a. There is a variable corresponding to each of the n2 positions on the board.

There is a variable corresponding to each knight.

b. Each variable can take one of two values, {occupied, vacant}

  Each variable’s domain is the set of squares.

c. every pair of squares separated by a knight’s move is constrained, such that both cannot be occupied. Furthermore, the entire set of squares is constrained, such that the total number of occupied squares should be k.

  every pair of knights is constrained, such that no two knights can be on the same square or on squares separated by a knight’s move. Solution B may be preferable because there is no global constraint, although Solution A has the smaller state space when k is large.

d. We will solve this using local search. Briefly describe in English a sensible successor function. Any solution must describe a complete-state formulation because we are using a local search algorithm. For simulated annealing, the successor function must completely connect the space; for random-restart, the goal state must be reachable by hill climbing from some initial state. Two basic classes of solutions are:
ensure no attacks at any time. Actions are to remove any knight, add a knight in any un attacked square, or move a knight to any un attacked square.
allow attacks but try to get rid of them. Actions are to remove any knight, add a knight in any square, or move a knight to any square.

the number of knights placed on the board. Since all states have no attacks, the global optimum of this function is in fact the optimal solution.
Here we need to penalize for attacks. One might suggest maximizing #knights - #attacks, but one must be careful to avoid the possibility that the score can be improved by adding lots more knights at the cost of a few extra attacks. One can show that #knights - 2#attacks works.