What is column reduction in the Hungarian method and how is it applied to a 4x4 matrix?
Conditions
- The matrix must have already undergone row reduction.
- The problem is a balanced assignment problem.
Reasoning, step by step
- Take the row-reduced matrix.
- Identify the minimum value in the first column.
- Subtract this minimum from all elements in the first column.
- Repeat for the second, third, and fourth columns.
- Assemble the transformed columns into the new matrix.
Example
In the row-reduced matrix, column 2 has entries [2, 5, 3, 4]. The minimum is 2. Subtracting 2 from each entry gives [0, 3, 1, 2].
Common misconceptions
- Applying column reduction to the original cost matrix instead of the row-reduced matrix.
- Subtracting the row minimum instead of the column minimum.
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Related questions
To select independent zeros, scan the reduced matrix for rows or columns with only one available zero. Circle that zero and strike out other zeros in the same row and column.
Conditions: The matrix has been reduced using row and column operations.; The goal is to find n independent zeros for an n x n matrix.
The presenter avoids selecting a zero when a row has two zeros because the goal is to find a set of independent zeros where each row and column has exactly one selected zero. Choosing arbitrarily from a row with multiple zeros might lead to conflicts later, so the process is postponed until the zero pattern is more constrained.
Conditions: The matrix has undergone row and column reduction.; The goal is to find an optimal assignment.
A balanced assignment problem means that the number of rows (persons) is equal to the number of columns (jobs). This allows for a one-to-one assignment where each person is assigned to exactly one job.
Conditions: The cost matrix is square (same number of rows and columns).; Each job is to be assigned to exactly one person.
To calculate the total cost, you must look at the original cost matrix, not the reduced matrix. Identify the positions of the circled zeros in the reduced matrix, find the corresponding entries in the original matrix, and sum those values.
Conditions: The optimal assignment has been determined from the reduced matrix.; The original cost matrix is available.
The purpose of step three is to inspect the reduced matrix row by row and count how many zeros each row contains. This helps determine if a unique assignment can be made or if further steps are needed to isolate independent zeros.
Conditions: The matrix has undergone row and column reduction.; The goal is to find an optimal assignment.
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