0/1 Knapsack Problem

Definition

You have a bag (Knapsack) with a weight limit $W$. You have a list of items, each with a weight and a value. You want to maximize the total value in the bag without exceeding the weight limit. You cannot break items (0/1 property: either take it or leave it).

How it Works

For every item, you have a choice:

1. Include it: Add its value to the total, subtract its weight from capacity.
2. Exclude it: Skip it and keep the capacity the same.
3. Result: max(Include, Exclude).

Code Example (Java)

def knapSack(W, wt, val, n):
    dp = [[0 for x in range(W + 1)] for x in range(n + 1)]

    # Build table dp[][] in bottom up manner
    for i in range(n + 1):
        for w in range(W + 1):
            if i == 0 or w == 0:
                dp[i][w] = 0
            elif wt[i-1] <= w:
                # Max of (including item, excluding item)
                dp[i][w] = max(val[i-1] + dp[i-1][w-wt[i-1]],  dp[i-1][w])
            else:
                dp[i][w] = dp[i-1][w]

    return dp[n][W]

val = [60, 100, 120]
wt = [10, 20, 30]
W = 50
n = len(val)
print(knapSack(W, wt, val, n)) # Output: 220

Complexity Analysis

• Time Complexity: O(N times W) where N is number of items and W is capacity.
• Space Complexity: O(N times W.

Use Cases

• Resource Allocation: Budgeting money for projects.
• Cargo Loading: Optimizing truck/ship loads.