How does the Hungarian algorithm work in finding the optimal solution for the assignment problem?
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The Hungarian algorithm is a combinatorial optimization technique that solves the assignment problem in polynomial time. The goal of the algorithm is to find the optimal assignment that minimizes the total cost or maximizes the total profit. Here are the steps of the Hungarian algorithm:
- Create a matrix of the costs or profits associated with each possible assignment.
- Subtract the smallest value in each row from all the values in that row, and then subtract the smallest value in each column from all the values in that column.
- Draw the minimum number of horizontal and vertical lines needed to cover all the zeros in the matrix. If the number of lines equals the number of rows or columns, an optimal solution has been found. If not, proceed to step 4.
- Find the smallest uncovered value in the matrix, and subtract it from all the uncovered values. Then, go back to step 3.
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The Hungarian algorithm efficiently finds the optimal assignment in the assignment problem by iteratively adjusting the cost matrix and selecting the minimal number of lines necessary to cover all zeros, yielding the optimal assignment configuration.
The Hungarian algorithm helps you figure out the easiest way or best way to assign tasks to workers in the workplace, by adjusting a chart of costs or the benefits, drawing lines to cover all zeros, and repeating this process until you find the best optimal assignments.