Question

Consider the Simplified Knapsack Problem: Suppose there are n objects with positive integer weights ai​i=1,…,n. Given a "knapsack" with capacity C, is it possible to fill up the knapsack exactly? In other words, does there exist a subset S⊆{1,…,n} such that ∑i∈S​ai​=C. Give an algorithm based on dynamic programming, and discuss its run-time.

Answer 1

The Knapsack Problem is a classic optimization problem that involves selecting a subset of objects to maximize total weight within a given capacity. A dynamic programming algorithm can solve this problem in polynomial time complexity.

Step-by-step explanation:

The Knapsack Problem is a classic optimization problem in computer science and mathematics. It involves selecting a subset of objects that maximize the total sum of their weights, while staying within the constraints of a given capacity. A dynamic programming approach can be used to solve this problem.

The algorithm involves creating a 2D table of size (n+1) x (C+1), where n is the number of objects and C is the capacity of the knapsack. Each cell (i, j) in the table represents the maximum weight that can be achieved using the first i objects and a knapsack with capacity j. The table can be filled in a bottom-up manner by considering each object and updating the table based on whether the current object can be included or not.

The runtime of the algorithm is O(nC), where n is the number of objects and C is the capacity of the knapsack. This is because for each object, we consider updating the table for each possible capacity value from 1 to C. Therefore, the algorithm runs in polynomial time complexity.