Question

Use the Backtracking algorithm for the \(0-1\) Knapsack problem (Algorithm 5.7) to maximize the profit for the following problem instance. Show the actions step by step. $$\begin{array}{ccccc} i & p_{i} & w_{i} & \frac{p_{i}}{w_{i}} & \\ 1 & \$ 20 & 2 & 10 & \\ 2 & \$ 30 & 5 & 6 & \\ 3 & \$ 35 & 7 & 5 & W=19 \\ 4 & \$ 12 & 3 & 4 & \\ 5 & \$ 3 & 1 & 3 & \end{array}$$

Short answer

The maximum profit that can be attained by using the Backtracking algorithm in a 0-1 Knapsack problem with a weight limit of 19 and given weights and profits of items is $100. This is achieved by including 1st, 2nd, and either 3rd or 4th item in the knapsack whenever possible.

Step-by-step solution

Initialize Variables

Firstly, the profit and weight of items are initialized. According to the given exercise, the weights are \[w = \{2, 5, 7, 3, 1\}\] and the profits are \[p = \{20, 30, 35, 12, 3\}\]. The maximum weight of the knapsack (\(W\)) is given as 19. The maximumProfit is initialized to 0 and nodeProfit is calculated as the sum of all given profits which is 100.

Start with First Item

The backtracking process begins with the first item. A decision is made whether to include the first item in the knapsack or not. Two scenarios are considered - first is by including the item and the second one is without including the item.

Calculate Profits

In the first scenario, when the item is included, the nodeWeight is added to the weight of the current item and nodeProfit is subtracted by the value of the current item. If this included item doesn't cause the total weight to exceed \(W\), then it's a promising node. The maximumProfit is updated

Backtrack if Necessary

In the second scenario, if the total weight exceeds \(W\), then backtrack and try the next item. This can be achieved by not adding the weight of the current item (i.e., the weight is considered as 0) and repeating Step 3.

Repeat Process

The process is repeated for all the items in the list. At the end of the process, the maximumProfit contains the maximum profit that can be achieved.

Key concepts

0-1 Knapsack Problem

The 0-1 Knapsack Problem is a classic example of an optimization problem where we aim to maximize the total profit without exceeding a maximum weight limit. You are given a set of items, each with a weight and a profit. The aim is to decide which items to include in a knapsack to get the highest possible profit, while ensuring that the total weight does not exceed a given limit. Unlike the fractional knapsack problem, you cannot break items into smaller parts. You must either take an item as a whole (denoted as "1") or leave it out completely (denoted as "0"). This is where the '0-1' terminology comes from. The problem is often solved using algorithms like Dynamic Programming or Backtracking, which test multiple combinations to find the optimal solution.

Algorithm Design

Algorithm design is crucial in solving problems like the 0-1 Knapsack Problem. It involves creating a strategy or a sequence of steps to transform inputs into the desired outputs. The main goal is to develop an efficient solution that uses less computational resources and time. Backtracking is one of the algorithm design strategies that particularly shines with combinatorial problems.

Backtracking incrementally builds candidates for solutions and abandons them if they are not suitable. This makes it particularly suited for problems where multiple solution paths exist, allowing it to discard paths that don’t meet the solution criteria quickly.

• Initialize necessary variables, such as current weight and profit.
• Begin exploring solutions from the first available item.
• Iteratively decide to include or exclude items based on constraints.
• Backtrack whenever constraints are violated to search alternative paths.

This structured method helps in efficiently navigating through the possible choices, ensuring all potential solutions are considered.

Profit Maximization

Profit maximization in the context of the 0-1 Knapsack Problem involves selecting a combination of items that maximizes the profit without exceeding the weight limit. The challenge is to make sure that each item included contributes the best possible value.

To accomplish this, every decision to include an item considers:

• The item's contribution to total profit.
• Remaining weight capacity in the knapsack.
• Potential profits of other items still not considered.

This delicate balancing act is essential for reaching the optimal solution. By tracking the current and potential maximum profits at each stage, a decision can be made whether to include an item or to temporarily "skip" it and try other combinations. By the end of the algorithm, the decision path that delivers the highest profit determines the chosen items.

Combinatorial Optimization

Combinatorial Optimization is concerned with optimizing an objective function over a finite set of feasible solutions. In the case of the 0-1 Knapsack Problem, the task is to identify the best combination of items. This often involves weighing numerous possible item sets against each other, taking into account both profit and weight constraints.

Using a Backtracking algorithm supports tackling this problem by:

• Evaluating subsets of items to form the best collection.
• Developing an approach to systematically explore all potential item combinations.
• Pruning away combinations that are not feasible based on weight limitations.

The goal is to find the most beneficial arrangement of items that maximizes the profit. By recursively exploring and evaluating sets of items, Backtracking ensures that no potential solution is overlooked, and the optimal solution can be discovered efficiently.