Question

Consider the following more generConsider the following more general version of the Knapsack problem. There are p groups of objects O1,O2,. . . ,Op and a knapsack capacity W. Each object x has a weight wx and a value vx. Our goal is to select a subset of objects such that: • the total weights of selected objects is at most W, • at most one object is selected from any group, and • the total value of the selected objects is maximized. Suppose that n is the total number of objects in all the groups Give an O(nW) time algorithm for this general Knapsack problem. Explain why your algorithm is correct and analyze the running time of your algorithm. Al version of the Knapsack problem. There are p groups of objects O1,O2,. . . ,Op and a knapsack capacity W. Each object x has a weight wx and a value vx. Our goal is to select a subset of objects such that: • the total weights of selected objects is at most W, • at most one object is selected from any group, and • the total value of the selected objects is maximized. Suppose that n is the total number of objects in all the groups Give an O(nW) time algorithm for this general Knapsack problem. Explain why your algorithm is correct and analyze the running time of your algorithm

Answer 1

Here is an O(nW) time algorithm for the generalized knapsack problem:

1. Create a 2D array dp of size (n+1) x (W+1). dp[i][w] will store the maximum value that can be achieved with weight <= w using objects up to i.

2. Initialize dp[0][w] = 0 for all w, as no objects are considered yet.

3. For each object i:

- For each weight w from 0 to W:

- If the weight of object i (wi) is <= w, dp[i][w] = max(dp[i-1][w], dp[i-1][w-wi] + vi)

- Else, dp[i][w] = dp[i-1][w]

4. Return dp[n][W]

This implements a bottom-up dynamic programming approach. We consider each object and fill in the dp table by either not picking the object (dp[i-1][w]) or picking it if it fits (dp[i-1][w-wi] + vi).

Correctness: The dp array stores the optimal solutions for all prefix sets of objects and all possible weight capacities. By building it up from empty to full object sets, we iteratively find the best solutions.

Time complexity:

- There are n objects and W possible weights, so the 2D array is of size n*W.

- Each entry takes O(1) work to compute.

- Therefore, the overall time complexity is O(n*W).

So this solves the generalized knapsack problem in O(nW) time.