How can you use the Hungarian algorithm to solve assignment problems?

2 Algorithm example

The Hungarian algorithm provides a way to determine the optimal assignment of workers to machines. Let's explore how it works with a simple example. Suppose we have four workers and four machines, and the cost matrix is as follows: 9, 2, 7, 8 for M1; 6, 4, 3, 7 for M2; 5, 8, 1, 8 for M3; and 7, 6, 9, 4 for M4. To assign each worker to one machine and minimize the total cost, we must subtract the smallest element in each row from every element in that row. Then we subtract the smallest element in each column from every element in that column. Next we cover all the zeros in the matrix with a minimum number of horizontal and vertical lines. If the number of lines is equal to the size of the matrix, then the optimal solution is found; otherwise we modify the matrix by adding or subtracting some constant to uncovered and covered elements respectively and repeat from step 3. In this case, the number of lines is equal to the size of the matrix so we have found the optimal solution. We can identify the optimal assignments by finding a set of independent zeros. One possible set of independent zeros is marked with asterisks: W1 to M2; W2 to M3; W3 to M4; and W4 to M1. The total cost of these assignments is 24 - the lowest possible cost for this problem.

3 Algorithm complexity

The Hungarian algorithm is a polynomial-time algorithm, meaning that it can solve any assignment problem in a reasonable amount of time, depending on the size of the problem. The exact complexity of the algorithm depends on the implementation, but it is generally bounded by O(n^3) or O(n^4), where n is the number of agents or tasks. This is much faster than trying to enumerate all possible assignments, which would take O(n!) time.

4 Algorithm applications

The Hungarian algorithm is a powerful tool that can be used to solve a range of real-world problems which involve matching agents to tasks. Examples include scheduling workers to shifts, assigning drivers to routes, pairing students with projects, and finding stable matches between partners. It takes into account availability, preferences, skills, distance, cost, convenience, interest, expertise and compatibility when making the assignments.

5 Algorithm limitations

The Hungarian algorithm is a powerful and efficient tool for solving assignment problems; however, it has some limitations. For instance, it requires a square or rectangular cost matrix which may not always be available or realistic. Additionally, it assumes the objective function is linear and additive, meaning the cost of each assignment does not depend on the other assignments, and the total cost is simply the sum of individual costs. This may not be true for some problems where the cost of each assignment may vary depending on the context or other constraints or criteria to consider. Furthermore, it finds only one optimal solution which may not be unique or preferable. For example, there may be multiple ways to assign agents to tasks with the same total cost but some may be more fair, balanced, or diverse than others. Alternatively, there may be no optimal solution but only a trade-off between different objectives such as cost, quality, or satisfaction.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?