Question

Suppose we assign \(n\) persons to \(n\) jobs. Let \(C_{U}\) be the cost of assigning the ith person to the jth job. Use a greedy approach to write an algorithm that finds an assignment that minimizes the total cost of assigning all \(n\) persons to all \(n\) jobs. Analyze your algorithm, and show the results using order notation.

Short answer

The greedy algorithm works by choosing the minimum cost job available at each step, which leads to an overall minimum cost. The cost matrix, deletion of rows and columns, and calculation of final cost are significant steps in this process. The time and space complexity of the algorithm is \(O(n^{2})\).

Step-by-step solution

Initialize Data

Start by creating a matrix where each entry \(C_{ij}\) represents the cost of assigning the ith person to the jth job.

Apply Greedy Algorithm

Choose the cheapest cost possible for assigning the job from the matrix. Once a job is assigned by selecting the minimum cost, remove that job from the list of available jobs.

Iterate over the Jobs

Repeat Step 2 for each remaining job. Each time, selecting the job with the minimum cost that has not yet been assigned. Delete the row corresponding to the person and the respective column for the job from the matrix once assigned.

Calculate the Overall Cost

After all jobs are assigned, calculate the total cost of assignment by adding up the selected costs.

Algorithm Analysis

Analyze the algorithm's efficiency. This algorithm has a time complexity of \(O(n^2)\) as we are making \(n\) selections, and each selection may involve scanning up to \(n\) elements. For space complexity, since we are storing the cost matrix that is also \(O(n^2)\)

Key concepts

Assignment Problem

The assignment problem is a fundamental topic in combinatorial optimization. It involves finding the most cost-effective way to assign a set number of agents, like people or machines, to a set number of tasks or jobs. Here, the challenge is to minimize the total assignment cost by pairing each person to a distinct job. For example, in a company setting, an assignment problem might determine which employee is best suited for a specific project to minimize expenses.
The assignment problem is usually represented by a cost matrix, where each element indicates the cost of assigning a particular person to a specific task. A crucial feature of this problem is that each job can only be completed by one person, and each person can only be assigned one job, ensuring a one-to-one mapping.
Using a greedy algorithm for solving this problem involves choosing the best immediate option without worrying about whether it will lead to the optimal global solution. This means consistently selecting the job assignment with the current minimum cost until all jobs are assigned.

Time Complexity

Time complexity is an essential consideration when assessing an algorithm's performance. It provides an estimate of the time an algorithm might take to complete in terms of the size of the input, usually denoted as \(n\). In our greedy approach to solving the assignment problem, the algorithm's time complexity is \(O(n^2)\). This notation implies that time taken increases quadratically with the increase in assignments and jobs.

• The algorithm checks each possible assignment to find the smallest cost for each iteration.
• As each job is assigned, we remove that option from the pool, effectively reducing the problem size incrementally.
• The complexity arises due to potentially needing to scan each remaining element of the matrix at each step, which leads to \(n\) iterations with \(n\) possible comparisons.

Although a \(O(n^2)\) complexity might seem feasible for smaller matrices, it can become inefficient for extremely large datasets.
Thus, while the greedy algorithm we use is quick and easy to implement, it may not always be the best choice for extremely large-scale problems.

Matrix Representation

Matrix representation is a powerful tool used to depict assignment problems distinctly. It allows for a clear visualization of the cost relationships between agents and tasks. In a matrix for our problem:

• Rows represent the agents, i.e., people, who need to be assigned jobs.
• Columns signify the tasks or jobs available for allocation.
• Each cell \(C_{ij}\) represents the cost associated with assigning the \(i\)-th person to the \(j\)-th job.

Such a structure provides an intuitive way to apply greedy algorithms.
The algorithm systematically examines the matrix to find the least costly assignment available. After selecting the cheapest option, it removes that row and column, effectively choosing a unique pair and reducing the matrix's dimension until all jobs are uniquely assigned.
This reduction process mimics crossing out options to visually depict elimination, allowing us to keep track of decisions and simplify the optimization problem dynamically as the algorithm progresses.

Cost Minimization

Cost minimization is the primary goal in solving the assignment problem efficiently. The purpose is to ensure that the total cost incurred from all assignments is the lowest possible. Each decision made by the algorithm should contribute to reducing the overall expenditure. Through this specific greedy algorithm:

• We single out the minimum cost for each unique job assignment from those available.
• By iteratively reducing choices, the algorithm minimizes potential wastage of resources.
• Ultimately, the total assignment cost is calculated by adding these minimized costs together.

The greedy strategy may not always lead to the global optimal solution in all scenarios, but it often achieves a fairly low-cost allocation effectively and swiftly.
In real-world applications, especially where time or computational resources are limited, a near-optimal yet efficient solution through cost minimization using greedy approaches becomes invaluable.