In the dynamic programming solution without item limit you have 2D matrix where Y-axis is item index and X-axis is weight. Then for each item, weight pair you choose maximum between
- value of weight including item if item weight <= weight limit
- value of weight excluding item
Here's example of the standard solution in Python:
def knapsack(n, weight, values, weights):
dp = [[0] * (weight + 1) for _ in range(n + 1)]
for y in range(1, n + 1):
for x in range(weight + 1):
if weights[y - 1] <= x:
dp[y][x] = max(dp[y - 1][x],
dp[y - 1][x - weights[y - 1]] + values[y - 1])
else:
dp[y][x] = dp[y - 1][x]
return dp[-1][-1]
Now when you add the item limit you have to choose maximum value for each item, value, number of items used triplet from
- value of weight and n items including item if item weight <= weight limit
- value of weight and n items excluding item
In order to represent number of items you can just add third dimension to the previously used matrix that represents the number of used items:
def knapsack2(n, weight, count, values, weights):
dp = [[[0] * (weight + 1) for _ in range(n + 1)] for _ in range(count + 1)]
for z in range(1, count + 1):
for y in range(1, n + 1):
for x in range(weight + 1):
if weights[y - 1] <= x:
dp[z][y][x] = max(dp[z][y - 1][x],
dp[z - 1][y - 1][x - weights[y - 1]] + values[y - 1])
else:
dp[z][y][x] = dp[z][y - 1][x]
return dp[-1][-1][-1]
Simple demo:
w = 5
k = 2
values = [1, 2, 3, 2, 2]
weights = [4, 5, 1, 1, 1]
n = len(values)
no_limit_fmt = 'Max value for weight limit {}, no item limit: {}'
limit_fmt = 'Max value for weight limit {}, item limit {}: {}'
print(no_limit_fmt.format(w, knapsack(n, w, values, weights)))
print(limit_fmt.format(w, k, knapsack2(n, w, k, values, weights)))
Output:
Max value for weight limit 5, no item limit: 7
Max value for weight limit 5, item limit 2: 5
Note that you could optimize the example a bit regarding memory consumption since when adding zth item to the solution you only need to know the solution for z - 1 items. Also you could check if it's is possible to fit z items under the weight limit to begin with and if not reduce the item limit accordingly.
