scipy.optimize.linear_sum_assignment

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4.5.4.4. scipy.optimize.linear_sum_assignment ¶

Solve the linear sum assignment problem.

The linear sum assignment problem 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

s.t. each row is assignment to at most one column, and each column to at most 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.

The method used is the Hungarian algorithm, also known as the Munkres or Kuhn-Munkres algorithm.

Parameters:	: array
Returns:	: array col_ind].sum(). The row indices will be sorted; in the case of a square cost matrix they will be equal to .

Parameters:

: array

Returns:

: array

col_ind].sum(). The row indices will be sorted; in the case of a square cost matrix they will be equal to .

New in version 0.17.0.

http://www.public.iastate.edu/~ddoty/HungarianAlgorithm.html
Harold W. Kuhn. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly , 2:83-97, 1955.
Harold W. Kuhn. Variants of the Hungarian method for assignment problems. Naval Research Logistics Quarterly , 3: 253-258, 1956.
Munkres, J. Algorithms for the Assignment and Transportation Problems. J. SIAM , 5(1):32-38, March, 1957.
https://en.wikipedia.org/wiki/Hungarian_algorithm

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Optimization in SciPy

SciPy is a Python library that is available for free and open source and is used for technical and scientific computing. It is a set of useful functions and mathematical methods created using Python’s NumPy module.

Features of SciPy:

Creating complex programs and specialized applications is a benefit of building SciPy on Python.
SciPy contains varieties of sub-packages that help to solve the most common issue related to Scientific Computation.
SciPy users benefit from the inclusion of new modules created by programmers all around the world in a variety of software-related fields.
Easy to use and understand as well as fast computational power.
It can operate on an array of NumPy libraries.

Sub-packages of SciPy:


	File input/output
	Special Function(airy, elliptic, bessel, gamma, beta, etc)
	Linear Algebra Operation
	Interpolation
	Optimization and fit
	Statistics and random numbers
	Numerical Integration
	Fast Fourier transforms
	Signal Processing
	Image manipulation

In this article, we will learn the scipy.optimize sub-package.

This package includes functions for minimizing and maximizing objective functions subject to given constraints. Let’s understand this package with the help of examples.

SciPy – Root Finding

func : callable The function whose root is required. It must be a function of a single variable of the form f(x, a, b, c, . . . ), where a, b, c, . . . are extra arguments that can be passed in the args parameter. x0 : float, sequence, or ndarray Initial point from where you want to find the root of the function. It will somewhere near the actual root. Our initial guess is 0.

Import the optimize.newton package using the below command. Newton package contains a function that will calculate the root using the Newton Raphson method . Define a function for the given objective function. Use the newton function. This function will return the result object which contains the smallest positive root for the given function f1.

SciPy – Linear Programming

Maximize : Z = 5x + 4y Constraints : 2y ≤ 50 x + 2y ≤ 25 x + y ≤ 15 x ≥ 0 and y ≥ 0

Import the optimize.linprog module using the following command. Create an array of the objective function’s coefficients. Before that convert the objective function in minimization form by multiplying it with a negative sign in the equation. Now create the two arrays for the constraints equations. One array will be for left-hand equations and the second array for right-hand side values.

Use the linprog inbuilt function and pass the arrays that we have created an addition mention the method.

This function will return an object that contains the optimal answer for the given problem.

SciPy – Assignment Problem

A city corporation has decided to carry out road repairs on main four arteries of the city. The government has agreed to make a special grant of Rs. 50 lakh towards the cost with a condition that repairs are done at the lowest cost and quickest time. If the conditions warrant, a supplementary token grant will also be considered favourably. The corporation has floated tenders and five contractors have sent in their bids. In order to expedite work, one road will be awarded to only one contractor. Cost of Repairs ( Rs in lakh) Contractors R1 R2 R3 R4 R5 C1 9 14 19 15 13 C2 7 17 20 19 18 C3 8 18 21 18 17 C4 10 12 18 19 18 C5 10 15 21 16 15 Find the best way of assigning the repair work to the contractors and the costs. If it is necessary to seek supplementary grants,what should be the amount sought?

In this code, we required the NumPy array so first install the NumPy module and then import the required modules. Create a multidimensional NumPy array of given data. Use the optimize.linear_sum_assignment() function. This function returns two NumPy arrays (Optimal solution) – one is the row ( Contactors) and the second is the column ( Corresponding Repair Cost).

Use the sum() function and calculate the total minimum optimal cost.

How the assignment will be done can be concluded below from the obtained data.

Contractor C1 will assign road 1 with a repair cost of 13Rs Lakh.
Contractor C2 will assign road 2 with a repair cost of 7RS lakh.
Contractor C3 will assign road 3 with a repair cost of 21RS lakh.
Contractor C4 will assign road 4 with a repair cost of 12RS lakh.
Contractor C5 will assign road 5 with a repair cost of 16RS lakh.

Hence Final Minimal cost will be,

SciPy – Minimize

Broyden-fletcher-goldfarb-shanno ( bfgs ).

This algorithm deals with the Minimization of a scalar function of one or more variables. The BFGS algorithm is one of the most widely used second-order algorithms for numerical optimization, and it is frequently used to fit machine learning algorithms such as the logistic regression algorithm.

Objective Function: z = sin( X ) + cos( Y )

SciPy – curve_fit

Curve fitting requires that you define the function that maps examples of inputs to outputs like Machine Learning supervised learning . The mapping function can have a straight line ( Linear Regression ), a Curve line ( Polynomial Regression ), and much more. A detailed GeeksForGeeks article on Scipy Curve_fit is available here SciPy | Curve Fitting .

Approximate curve fitted to the input points

SciPy – Univariate Function Minimizers

Non-linear optimization with no constraint and there is only one decision variable in this optimization that we are trying to find a value for.

Nelder-Mead Simplex Search – Machine Learning

Noisy Optimization Problem – A noisy objective function is a function that gives different answers each time the same input is evaluated.

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Is there an algorithm for solving a many-to-one assignment problem?

I'm looking for an algorithm to solve assignment problem, but it is not one to one problem. I know the Hungarian algorithm it can simply assign one task to one agent.

Let's say I have 4 tasks and 20 agents and I want to assign 5 agents to each task (based on the cost matrix). Is there any efficient algorithm to do this?

It will be great if there is a Python library with algorithm like that.

optimization

2 Answers 2

Let's say that you now have a 20 by 4 cost matrix $C$ consisting of the costs of assigning agents to tasks. You can make 5 new tasks, each requiring one agent, out of each original task.

To do this, make a new cost matrix $C_{\text{new}}$ , which is 20 by 20, in which each column of $C$ appears 5 times. Use the Hungarian algorithm on $C_{\text{new}}$ . and you will have 1 agent assigned to every new task, which will therefore be 5 agents assigned to every original task.

Using the matrix scheme suggested by Mark, you could use the Jonker-Volgenant algorithm which is a modification of the Hungarian algorithm.

It is implemented in scipy.optimize.linear_sum_assignment . Here is an example from the documentation, which you can modify to include your own choice of cost matrix.

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Linear sum assignment, but with ranked assignments?

Let's say I have 5 tasks that I have to assign to 5 agents, with a cost matrix:

Using scipy's optimization library, I can get assignments for each agent:

However, what if I'm only interested in the agents to assign task 0? Let's say I just want to understand for task 0 only a ranked list of agents to assign to that task, but still with the goal of minimizing total system-wide cost. Is there any way to solve for this problem?

linear-programming
combinatorial-optimization
assignment-problem

2 $\begingroup$ What about maybe solving the LAP n(umber agents) times, while each iteration fixes a different agent to task 0. Rank by objective-values after. I guess such partial-fixing should be trivial given a LP-based LAP-solver (a variable-bound of [1,1] instead of [0,1] doesn't destroy any relevant structure, e.g. total unimodularity). (One could even exploit warm-start / incrementality to speed it up i guess) $\endgroup$ – sascha Commented May 21 at 1:34
1 $\begingroup$ @sascha Ended up going with this approach! Added benefit is that I was able to run iterations in parallel thereby reducing runtime. $\endgroup$ – Dan Commented May 31 at 17:21

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Benchmarking Linear Assignment Problem Solvers

berhane/LAP-solvers

Folders and files.

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The script benchmarks the performance of Python3 linear assignment problem solvers for random cost matrices of different sizes. These solvers are:

https://github.com/scipy/scipy/
https://github.com/bmc/munkres
does not work with Python 3.6 and 3.7
https://github.com/Hrldcpr/Hungarian
https://github.com/gatagat/lap In addition, these two solvers are added for Python3
Please see the blog post here
https://github.com/src-d/lapjv
Please note that Christioph has also done a benchmark of LAP solvers
https://github.com/cheind/py-lapsolver
https://github.com/jdmoorman/laptools

They all formally have O(n 3 ) complexity, but their performance differs substantially based on their implementation and the size of the matrix they are trying to solve. The solvers can be classified based on some unique characteristics.

Module	Python or C/C++/Cython	Algorithm
scipy.optimize.linear_sum_assignment	Python(<v1.4)/C++(=>v1.4))	Hungarian
munkres.Munkres	Python	Hungarian
laptools.clap	Python	?
hungarian.lap	C++	Hungarian
lap.lapjv	C++	Jonker-Volgenant
lapjv.lapjv	C++	Jonker-Volgenant
lapsolver.solve_dense	C++	shortest augmenting path

The purpose of this benchmarking exercise is to see which implementation performs best for a given matrix size. My interest is to use this information to improve the performance of Arbalign and expand its use.

The repo contains the following:

benchmark-lap-solvers.py - a Python3 script comparing four/six implementations
benchmark-lap-solvers-py3.ipynb - a Jupyter notebook comparing four/six implementations. It has been tested using Python 3.6 and 3.7.

It's simple once you have installed the necessary packages.

command	execution	note
		default
		default, except it looks at small matrices only
		default, except plotting is suppressed
		default, except it prints lowest cost for each method

If you want to add other solvers to the list, it should be easy to figure out what parts to update in the scripts.

Requirements

pip3 install numpy
conda install numpy
pip3 install matplotlib
conda install matplotlib
pip3 install scipy==1.4
conda install scipy
pip3 install munkres
conda install munkres
pip3 install hungarian
conda install -c psi4 hungarian
pip3 install lap
conda install lap
pip3 install lapjv
pip3 install lapsolver
conda install -c loopbio lapsolver
pip3 install laptools (Python 3.5+)

The script will produce output similar to what's shown below. Some things to note are:

The timings here corresponds to an average of three Python 3.5.6 runs on CentOS 7 machine with 2.4 GHz Intel Xeon Gold 6148 processor and 192GB of RAM
The random matrices are filled with floating point numbers ranging from 0 to the size (# of rows or columns) of the matrix. They are generated using numpy: cost_matrix = matrix_size * np.random.random((matrix_size, matrix_size))
Data of timing for solving LAP of random cost matrices of sizes 2 min x 2 min to 2 max x 2 max .
plot of timing for LAP solving random cost matrices of sizes 2 min x 2 min to 2 max x 2 max , where min and max are limited to smaller numbers for munkres and scipy in the interest of time.

If requested via the --printcost flag, it will also print the minimum cost for each random cost matrix by each implementation. This test ensures that the methods are making consistent/correct assignments.

scipy==1.4 is much faster than previous versions and it is competitive with the other implementations, especially for larger matrices. This is a great development since it probably gets used more than the other implementations by virtue of scipy's popularity.
munkres is much slower than hungarian , lapsolver , scipy , lap.lapjv , and lapjv.lapjv for all matrix sizes
hungarian performs well for smaller matrices. For anything larger than 256x256, lapsolver , lap.lapjv and lapjv.lapjv are about an order of magnitude faster than hungarian
lap.lapjv is am implementation intended to solve dense matrices. Its sparse matrix solver analog named lap.lapmod is more efficient for larger sparse matrices. Both are implemented in the lap module.
lapjv.lapjv has the best performance virtually for all matrix sizes.
For the purposes of improving Arbalign , hungarian remains a good choice for most molecular systems I'm interested in which don't have more than 100x100 distance matrices the same type to solve. However, if the tool is to be applied to larger molecules such as proteins and DNA, it would be worthwhile to use lapjv.lapjv , lapsolver , lap.lapjv or lap.lapmod
Jupyter Notebook 89.4%
Python 10.6%

linear_sum_assignment #

Solve the linear sum assignment problem.

The cost matrix of the bipartite graph.

Calculates a maximum weight matching if true.

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]) .

for sparse inputs

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

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

Added in version 0.17.0.

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

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

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Scipy's linear_sum_assignment giving incorrect result

When I tried using scipy.optimize.linear_sum_assignment as shown, it gives the assignment vector [0 2 3 1] with a total cost of 15.

However, from the cost matrix c , you can see that for the second task, the 5th agent has a cost of 1 . So the expected assignment should be [0 3 None 2 1] (total cost of 9)

Why is linear_sum_assignment not returning the optimal assignments?

optimization
linear-programming
hungarian-algorithm

linear_sum_assignment returns a tuple of two arrays. These are the row indices and column indices of the assigned values. For your example (with c converted to a numpy array):

The corresponding index pairs from row and col give the selected entries. That is, the indices of the selected entries are (0, 0), (1, 2), (3, 3) and (4, 1). It is these pairs that are the "assignments".

The sum associated with this assignment is 9:

In the original version of the question (but since edited), it looks like you wanted to know the row index for each column, so you expected [0, 4, 1, 3]. The values that you want are in row , but the order is not what you expect, because the indices in col are not simply [0, 1, 2, 3]. To get the result in the form that you expected, you have to reorder the values in row based on the order of the indices in col . Here are two ways to do that.

Note that the example in the linear_sum_assignment docstring is potentially misleading; because it displays only col_ind in the python session, it gives the impression that col_ind is "the answer". In general, however, the answer involves both of the returned arrays.

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