The key features of combinatorial problems
Combinatorial problems are problems that involve finding the number of ways to arrange or select objects from a given set, often under certain constraints. Some key features of combinatorial problems included:
- The "n-queens" problem: Place n queens on an nxn chessboard such that no two queens are attacking each other (i.e., no two queens are in the same row, column, or diagonal). Find the number of ways to do this.
- The "knapsack" problem: Given a list of items, each with a weight and a value, find the combination of items with the maximum total value that can be carried in a knapsack with a fixed weight limit.
- The "traveling salesman" problem: Given a list of cities and the distances between them, find the shortest possible route that visits each city exactly once and returns to the starting city.
Examples of combinatorial problems include:
- Constraints: There may be certain constraints on the arrangements or selections, such as the number of objects that can be chosen or the order in which the objects must be placed.
- Combinations or permutations: The problem may involve finding the number of combinations (unordered selections) or permutations (ordered arrangements) of the objects.
- A finite set of objects: There is a finite number of objects that can be arranged or selected in different ways.