scipy.optimize.linear_sum_assignment¶
- scipy.optimize.linear_sum_assignment(cost_matrix, maximize=False)[source]¶
Solve the linear sum assignment problem.
- Parameters
- cost_matrixarray
The cost matrix of the bipartite graph.
- maximizebool (default: False)
Calculates a maximum weight matching if true.
- Returns
- row_ind, col_indarray
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 tonumpy.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
- 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
>>> 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