How do you select independent zeros in the Hungarian method?
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.
Reasoning, step by step
- Identify a row or column with a single zero.
- Circle that zero.
- Strike out other zeros in the same row and column.
- Repeat for remaining rows and columns.
- Ensure all rows and columns have one circled zero.
Example
Row D has only one zero at column 1. Circle it and strike out other zeros in column 1. Then Row A has only one remaining zero at column 2, so circle it.
Common misconceptions
- Circling a zero without striking out conflicting zeros in the same row or column.
- Assuming that any set of zeros is valid, even if they share a row or column.
Watch the explanation
Connected concepts
Explore next
Related questions
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.
Column reduction is the second step of the Hungarian method. It involves taking the matrix obtained after row reduction, finding the minimum element in each column, and subtracting this minimum value from all elements in that same column.
Conditions: The matrix must have already undergone row reduction.; The problem is a balanced assignment problem.
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.
Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.