assignment hungarian method

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certainly! the hungarian method is an efficient algorithm used to solve the assignment problem, which aims to minimize the cost of assigning tasks to agents, where each task must be assigned to one agent and vice versa. this method is particularly useful for problems represented as a cost matrix.

problem definition

given a cost matrix `c` of size `n x n`, where `c[i][j]` represents the cost of assigning task `j` to agent `i`, the goal is to find a one-to-one assignment of agents to tasks that minimizes the total cost.

steps of the hungarian method

1. **row reduction**: subtract the smallest value in each row from all elements of that row.

2. **column reduction**: after row reduction, subtract the smallest value in each column from all elements of that column.

3. **cover zeros**: use a minimum number of lines to cover all zeros in the matrix. if the number of lines is equal to the number of rows (or columns), an optimal assignment exists among the zeros.

4. **adjust matrix**: if the number of lines is less than `n`, find the smallest uncovered value, subtract it from all uncovered elements, and add it to the elements that are covered twice.

5. **repeat**: go back to step 3 until an optimal assignment is found.

6. **construct the assignment**: once the optimal assignment is found, extract the assignment from the matrix.

example code in python

here's a python implementation of the hungarian method using the `scipy` library, which provides an efficient implementation of the algorithm:

```python
import numpy as np
from scipy.optimize import linear_sum_assignment

def hungarian_method(cost_matrix):
use scipy's linear_sum_assignment to find the optimal assignment
row_ind, col_ind = linear_sum_assignment(cost_matrix)
calculate the total cost of the optimal assignment
total_cost = cost_matrix[row_ind, col_ind].sum()

return row_ind, col_ind, total_cost

if __name__ == "__main__":
example cost matrix
cost_matrix = np.array([[4 ...

#HungarianMethod #AssignmentProblem #python
Hungarian method
assignment problem
linear programming
optimization algorithm
cost minimization
matrix representation
efficient allocation
combinatorial optimization
optimal assignment
transportation problem
algorithm efficiency
bipartite matching
matrix manipulation
decision-making process
resource allocation

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