Question

Suppose we have a solution to the \(n\) -Queens Problem instance in which \(n=4,\) Can we extend this solution to find a solution to the problem instance in which \(n=5,\) then use the solutions for \(n=4\) and \(n=5\) to construct a solution to the instance in which \(n=6,\) and continue this dynamic programming approach to find a solution to any instance in which \(n>4 ?\) Justify your answer.

Short answer

No, the solution to the n-Queens Problem for an instance \(n=4\) cannot be extended to solve for an instance \(n=5\), nor does this method serve as a dynamic programming approach to solve for \(n>4\). This is because the n-Queens Problem lacks optimal substructure, which is the property needed for a problem to be solved using the dynamic programming approach.

Step-by-step solution

Understand the n-Queens Problem

In the n-Queens Problem, the objective is to place \(n\) queens on an \(n \times n\) chessboard such that no two queens are under attack. A queen can attack any other queen along the row, column, or diagonal paths.

Analyze the Dynamic Programming Approach

The 'dynamic programming' way involves breaking down the problem into smaller sub-problems, solving these sub-problems, and using the solutions to solve the larger problem. However, in the n-Queens problem, the solution to an instance \(n=x\) cannot be used to directly solve for \(n=x+1\) (e.g., from \(n=4\) to \(n=5\) or \(n=5\) to \(n=6\)). This is because the number of queens and the size of the chessboard increase simultaneously.

Justify the Inapplicability of Dynamic Programming Approach

While it may seem immediately appealing to try to find a 'dynamic programming' solution for the n-Queens Problem, it won't work because the problem lacks optimal substructure. This means that an optimal solution to the n-Queens Problem cannot be constructed efficiently from optimal solutions of its sub-problems. Therefore, the solutions for \(n=4\) and \(n=5\) cannot be combined to create a solution for \(n=6\). Consequently, the use of dynamic programming is not a feasible strategy to find solutions for any instance \(n>4\) of the n-Queens Problem.