# scipy.optimize.linear_sum_assignment

scipy.optimize.linear_sum_assignment()

  Solve the linear sum assignment problem.

  Parameters:
    cost_matrix array

      The cost matrix of the bipartite graph.

    maximize bool (default: False)

      Calculates a maximum weight matching if true.

  Returns:
    row_ind, col_ind array

      An array of row indices and one of corresponding column indices giving the optimal assignment. The cost of the assignment can be computed as `cost_matrix[row_ind, col_ind].sum()`. The row indices will be sorted; in the case of a square cost matrix they will be equal to `numpy.arange(cost_matrix.shape[0])`.

  See also

  `scipy.sparse.csgraph.min_weight_full_bipartite_matching`

    for sparse inputs

  Notes

  The linear sum assignment problem [1] is also known as minimum weight matching in bipartite graphs. A problem instance is described by a matrix C, where each C[i,j] is the cost of matching vertex i of the first partite set (a ‘worker’) and vertex j of the second set (a ‘job’). The goal is to find a complete assignment of workers to jobs of minimal cost.

  Formally, let X be a boolean matrix where \(X[i,j] = 1\) iff row i is assigned to column j. Then the optimal assignment has cost

  \[\min \sum_i \sum_j C_{i,j} X_{i,j}\]

  where, in the case where the matrix X is square, each row is assigned to exactly one column, and each column to exactly one row.

  This function can also solve a generalization of the classic assignment problem where the cost matrix is rectangular. If it has more rows than columns, then not every row needs to be assigned to a column, and vice versa.

  This implementation is a modified Jonker-Volgenant algorithm with no initialization, described in ref. [2].

  New in version 0.17.0.

  References

  [1]

  https://en.wikipedia.org/wiki/Assignment_problem

  [2]

  DF Crouse. On implementing 2D rectangular assignment algorithms. IEEE Transactions on Aerospace and Electronic Systems, 52(4):1679-1696, August 2016, DOI:10.1109/TAES.2016.140952

  Examples

      >>> import numpy as np
      >>> cost = np.array([[4, 1, 3], [2, 0, 5], [3, 2, 2]])
      >>> from scipy.optimize import linear_sum_assignment
      >>> row_ind, col_ind = linear_sum_assignment(cost)
      >>> col_ind
      array([1, 0, 2])
      >>> cost[row_ind, col_ind].sum()
      5