The knapsack problem involves selecting items with given weights and values to maximize value within a weight limit. You can solve it using dynamic programming with a 2D array in TypeScript.

The knapsack problem is a classic optimization problem where you are given a set of items, each with a weight and a value, and a knapsack with a weight capacity. The goal is to determine the most valuable combination of items that can fit within the knapsack’s weight limit. This problem can be solved using dynamic programming (DP) by creating a 2D array where the rows represent the items and the columns represent the weight capacities from 0 to the maximum weight of the knapsack. Each cell in the table contains the maximum value that can be achieved with the current set of items and weight limit. The DP solution works by iterating through the items and for each item, deciding whether to include it in the knapsack or not. If the item is included, the value is updated by adding the item's value to the value of the remaining capacity (calculated from the previous row). If not, the value remains the same as in the previous row. This method has a time complexity of O(n * W), where n is the number of items and W is the weight capacity, making it efficient for moderate problem sizes. In TypeScript, you can implement the DP solution using a 2D array and nested loops to fill the table. The knapsack problem has practical applications in areas like resource allocation, financial investment planning, and cargo loading, where maximizing value within a limited capacity is critical.