Graph Theory Questions Long
The Hungarian algorithm, also known as the Kuhn-Munkres algorithm, is a combinatorial optimization algorithm used to solve the assignment problem in graph theory. The assignment problem involves finding the optimal assignment of a set of agents to a set of tasks, where each agent can only be assigned to one task and each task can only be assigned to one agent. The goal is to minimize the total cost or maximize the total benefit of the assignment.
The Hungarian algorithm is based on the concept of augmenting paths in bipartite graphs. It starts by initializing a matrix called the cost matrix, which represents the costs or benefits associated with each possible assignment. The algorithm then iteratively finds a series of augmenting paths in the graph until an optimal assignment is reached.
The algorithm consists of the following steps:
1. Step 1: Subtract the smallest element in each row from all the elements in that row. This step ensures that at least one zero is present in each row.
2. Step 2: Subtract the smallest element in each column from all the elements in that column. This step ensures that at least one zero is present in each column.
3. Step 3: Cover all the zeros in the matrix using the minimum number of lines. A line can be either a row or a column. The goal is to cover all the zeros using the minimum number of lines.
4. Step 4: If the number of lines used to cover the zeros is equal to the number of rows or columns, an optimal assignment has been found. If not, proceed to step 5.
5. Step 5: Find the smallest uncovered element in the matrix. Subtract this element from all the uncovered elements and add it to all the elements covered by two lines. Then, go back to step 3.
6. Step 6: Repeat steps 3 to 5 until an optimal assignment is found.
Once the algorithm terminates, the optimal assignment can be obtained by selecting the elements in the matrix that correspond to the zeros in the final matrix. The row and column indices of these zeros represent the assigned agents and tasks, respectively.
The Hungarian algorithm has a time complexity of O(n^3), where n is the number of agents or tasks. It is widely used in various applications such as job assignment, resource allocation, and scheduling problems.