This is a C++ Program to knapsack problem using dynamic programming. The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a mass and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.

Here is source code of the C++ Program to Solve Knapsack Problem Using Dynamic Programming. The C++ program is successfully compiled and run on a Linux system. The program output is also shown below.

`// A Dynamic Programming based solution for 0-1 Knapsack problem`

`#include <iostream>`

using namespace std;

`// A utility function that returns maximum of two integers`

int max(int a, int b)

`{`

return (a > b) ? a : b;

`}`

`// Returns the maximum value that can be put in a knapsack of capacity W`

int knapSack(int W, int wt[], int val[], int n)

`{`

int i, w;

int K[n + 1][W + 1];

`// Build table K[][] in bottom up manner`

for (i = 0; i <= n; i++)

`{`

for (w = 0; w <= W; w++)

`{`

if (i == 0 || w == 0)

K[i][w] = 0;

else if (wt[i - 1] <= w)

K[i][w]

= max(val[i - 1] + K[i - 1][w - wt[i - 1]], K[i - 1][w]);

`else`

K[i][w] = K[i - 1][w];

`}`

`}`

return K[n][W];

`}`

int main()

`{`

cout << "Enter the number of items in a Knapsack:";

int n, W;

cin >> n;

int val[n], wt[n];

for (int i = 0; i < n; i++)

`{`

cout << "Enter value and weight for item " << i << ":";

cin >> val[i];

cin >> wt[i];

`}`

`// int val[] = { 60, 100, 120 };`

`// int wt[] = { 10, 20, 30 };`

`// int W = 50;`

cout << "Enter the capacity of knapsack";

cin >> W;

cout << knapSack(W, wt, val, n);

return 0;

`}`

Output:

$ g++ DPKnapsack.cpp $ a.out Enter the number of items in a Knapsack:5 Enter value and weight for item 0:11 111 Enter value and weight for item 1:22 121 Enter value and weight for item 2:33 131 Enter value and weight for item 3:44 141 Enter value and weight for item 4:55 151 Enter the capacity of knapsack 300 99

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