assignment problem hungarian method code

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Divide & Conquer

Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)

Given a 2D array , arr of size N*N where arr[i][j] denotes the cost to complete the j th job by the i th worker. Any worker can be assigned to perform any job. The task is to assign the jobs such that exactly one worker can perform exactly one job in such a way that the total cost of the assignment is minimized.

Input: arr[][] = {{3, 5}, {10, 1}} Output: 4 Explanation: The optimal assignment is to assign job 1 to the 1st worker, job 2 to the 2nd worker. Hence, the optimal cost is 3 + 1 = 4. Input: arr[][] = {{2500, 4000, 3500}, {4000, 6000, 3500}, {2000, 4000, 2500}} Output: 4 Explanation: The optimal assignment is to assign job 2 to the 1st worker, job 3 to the 2nd worker and job 1 to the 3rd worker. Hence, the optimal cost is 4000 + 3500 + 2000 = 9500.

Different approaches to solve this problem are discussed in this article .

Approach: The idea is to use the Hungarian Algorithm to solve this problem. The algorithm is as follows:

Consider an example to understand the approach:

Let the 2D array be: 2500 4000 3500 4000 6000 3500 2000 4000 2500 Step 1: Subtract minimum of every row. 2500, 3500 and 2000 are subtracted from rows 1, 2 and 3 respectively. 0 1500 1000 500 2500 0 0 2000 500 Step 2: Subtract minimum of every column. 0, 1500 and 0 are subtracted from columns 1, 2 and 3 respectively. 0 0 1000 500 1000 0 0 500 500 Step 3: Cover all zeroes with minimum number of horizontal and vertical lines. Step 4: Since we need 3 lines to cover all zeroes, the optimal assignment is found. 2500 4000 3500 4000 6000 3500 2000 4000 2500 So the optimal cost is 4000 + 3500 + 2000 = 9500

For implementing the above algorithm, the idea is to use the max_cost_assignment() function defined in the dlib library . This function is an implementation of the Hungarian algorithm (also known as the Kuhn-Munkres algorithm) which runs in O(N 3 ) time. It solves the optimal assignment problem.

Below is the implementation of the above approach:

Time Complexity: O(N 3 ) Auxiliary Space: O(N 2 )

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Python 3 implementation of the Hungarian Algorithm

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Hungarian algorithm.

A Python 3 graph implementation of the Hungarian Algorithm (a.k.a. the Kuhn-Munkres algorithm), an O(n^3) solution for the assignment problem , or maximum/minimum-weighted bipartite matching problem .

The function find_matching takes 3 inputs:

*See examples below.

Example 1 (maximum-weighted matching)

Suppose you're choosing 11 starting positions for a soccer team.

The 11 players submit their top 3 position choices, and it is your job to create the optimal team.

The situation can be modeled with a weighted bipartite graph :

Then, if you assign weight 3 to blue edges, weight 2 to red edges and weight 1 to green edges, your job is simply to find the matching that maximizes total weight. This is the assignment problem , for which the Hungarian Algorithm offers a solution.

Notice: although no one has chosen LB, the algorithm will still assign a player there. In fact, the first step of the algorithm is to create a complete bipartite graph (all possible edges exist), giving new edges weight 0.

Define a weighted bipartite graph in the following manner:

thus defining a complete bipartite graph G = (L ∪ R, E) with:

Then pass the graph as an input:

You will get the output:

If you only need the total weight:

Example 2 (minimum-weighted matching)

Suppose you manage a group of drivers delivering packages to various locations.

You estimate the time of delivery for each driver to deliver each package, and it is your job to save the most time.

This time, we will model the situation with a matrix:

where the values in the matrix give the number of minutes it would take each driver to deliver each package. Again, this is the assignment problem , except this time we seek to find a minimum-weighted matching to minimize the total amount of delivery time.

Then pass the graph as an input (this time, remember to change matching_type = 'min' to find a minimum-weighted matching):

The algorithm was published by Harold Kuhn in 1955 paper The Hungarian Method for the Assignment Problem . Kuhn's work relied heavily on that of Hungarian mathematicians Dénes Kőnig and Jenő Egévary.

Contributors 2

Python 100.0%

Hungarian Algorithm Introduction & Python Implementation

How to use hungarian method to resolve the linear assignment problem..

By Eason on August 2nd, 2021

In this article, I will introduce how to use Hungarian Method to resolve the linear assignment problem and provide my personal Python code solution.

So… What is the linear assignment problem?

The linear assignment problem represents the need to maximize the available resources (or minimize the expenditure) with limited resources. For instance, below is a 2D matrix, where each row represents a different supplier, and each column represents the cost of employing them to produce a particular product. Each supplier can only specialize in the production of one of these products. In other words, only one element can be selected for each column and row in the matrix, and the sum of the selected elements must be minimized (minimized cost expense).

The cost of producing different goods by different producers:

Indeed, this is a simple example. By trying out the possible combinations, we can see that the smallest sum is 13, so supplier A supplies Bubble Tea , supplier B supplies milk tea, and supplier C supplies Fruit Tea . However, such attempts do not follow a clear rule and become inefficient when applied to large tasks. Therefore, the next section will introduce step by step the Hungarian algorithm, which can be applied to the linear assignment problem.

Hungarian Algorithm & Python Code Step by Step

In this section, we will show how to use the Hungarian algorithm to solve linear assignment problems and find the minimum combinations in the matrix. Of course, the Hungarian algorithm can also be used to find the maximum combination.

Step 0. Prepare Operations

First, an N by N matrix is generated to be used for the Hungarian algorithm (Here, we use a 5 by 5 square matrix as an example).

The above code randomly generates a 5x5 cost matrix of integers between 0 and 10.

If we want to find the maximum sum, we could do the opposite. The matrix to be solved is regarded as the profit matrix, and the maximum value in the matrix is set as the common price of all goods. The cost matrix is obtained by subtracting the profit matrix from the maximum value. Finally, the cost matrix is substituted into the Hungarian algorithm to obtain the minimized combination and then remapped back to the profit matrix to obtain the maximized sum value and composition result.

The above code randomly generates a 5x5 profit matrix of integers between 0 and 10 and generate a corresponding cost matrix

By following the steps above, you can randomly generate either the cost matrix or the profit matrix. Next, we will move into the introduction of the Hungarian algorithm, and for the sake of illustration, the following sections will be illustrated using the cost matrix shown below. We will use the Hungarian algorithm to solve the linear assignment problem of the cost matrix and find the corresponding minimum sum.

Example cost matrix:

Step 1. Every column and every row subtract its internal minimum

First, every column and every row must subtract its internal minimum. After subtracting the minimum, the cost matrix will look like this.

Cost matrix after step 1:

And the current code is like this:

Step 2.1. Min_zero_row Function Implementation

At first, we need to find the row with the fewest zero elements. So, we can convert the previous matrix to the boolean matrix(0 → True, Others → False).

Transform matrix to boolean matrix:

Corresponding Boolean matrix:

Therefore, we can use the “min_zero_row” function to find the corresponding row.

The row which contains the least 0:

assignment problem hungarian method code

Third, mark any 0 elements on the corresponding row and clean up its row and column (converts elements on the Boolean matrix to False). The coordinates of the element are stored in mark_zero.

Hence, the boolean matrix will look like this:

The boolean matrix after the first process. The fourth row has been changed to all False.

The process is repeated several times until the elements in the boolean matrix are all False. The below picture shows the order in which they are marked.

The possible answer composition:

Step 2.2. Mark_matrix Function Implementation

After getting Zero_mat from the step 2–1, we can check it and mark the matrix according to certain rules. The whole rule can be broken down into several steps:

Finally, the whole mark_matrx function is finished and then returns marked_zero , marked_rows , marked_cols. At this point, we will be able to decide the result based on the return information.

If we use the example cost matrix, the corresponding marked_zero , marked_rows, and marked_cols are as follows:

Step 3. Identify the Result

At this step, if the sum of the lengths of marked_rows and marked_cols is equal to the length of the cost matrix, it means that the solution of the linear assignment problem has been found successfully, and marked_zero stores the solution coordinates. Fortunately, in the example matrix, we find the answer on the first try. Therefore, we can skip to step 5 and calculate the solution.

However, everything is hardly plain sailing. Most of the time, we will not find the solution on the first try, such as the following matrix:

After Step 1 & 2 , the corresponding matrix, marked_rows, and marked_cols are as follows:

The sum of the lengths of Marked_Rows and Marked_Cols is 4 (less than 5).

Apparently, the sum of the lengths is less than the length of the matrix. At this time, we need to go into Step 4 to adjust the matrix.

Step 4. Adjust Matrix

In Step 4, we're going to put the matrix after Step 1 into the Adjust_Matrix function . Taking the latter matrix in Step 3 as an example, the matrix to be modified in Adjust_Matrix is:

The whole function can be separated into three steps:

Find the minimum value for an element that is not in marked_rows and not in marked_cols . Hence, we can find the minimum value is 1.

Subtract the elements which not in marked_rows nor marked_cols from the minimum values obtained in the previous step.

Add the element in marked_rows , which is also in marked_cols , to the minimum value obtained by Step 4–1.

Return the adjusted matrix and repeat Step 2 and Step 3 until the conditions satisfy the requirement of entering Step 5.

Step 5. Calculate the Answer

Using the element composition stored in marked_zero , the minimum and maximum values of the linear assignment problem can be calculated.

The minimum composition of the assigned matrix and the minimum sum is 18.

The maximum composition of the assigned matrix and the maximum sum is 43.

The code of the Answer_Calculator function is as follows:

The complete code is as follows:

Hungarian algorithm - Wikipedia

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The Assignment Problem (Using Hungarian Algorithm)

Ever encountered a problem where you wanted to divide work among people such that you do it in the most optimal way? Then this problem is just for you. The assignment problem assigns tasks to agents in the most optimal way! Read on ahead to know about the Assignment Problem and how to solve it using the “ Hungarian Algorithm ”.

What is the Assignment Problem?

Let there be n agents and n tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized. We need to ensure the following things:

Problem: In a construction site there are 4 cranes. Each crane must be allocated to one job. The time required for each job for each crane is shown in the table below. Find the best assignment of cranes to the jobs so that the time required to finish the jobs is minimum.

Applications of the Assignment Problem

Few applications of the Assignment Problem are:

Few Approaches to solve the Assignment Problem

The Hungarian Algorithm

1.Find the minimum element from each row and substract that value from all the elements of the row.

2. Find the minimum element from each column and substract that value from all the elements of the column.

3. Let m=minimum number of lines required to cover all the zeroes in the table

4. while(m!=number of row/columns)

5. Use the zeroes to assign possible combinations- i.e wherever there is a zero present, task can be assigned.

6. Find the minimum cost

Solving an Example Using Hungarian Problem

We use the previous problem statement (in the example mentioned above) and solve that problem using Hungarian Algorithm.

STEP 1: Finding the minimum from each row and substracting it from all the other row elements

STEP 2: Finding the minimum from each column and substracting it from all the other column elements

STEP 3: Finding the minimum number of lines required to cover all the zeroes

As we can see that when we draw lines on column 1 and 2 and row 3, all the zeroes of the table are covered but this is not equal to the number of rows and columns.

STEP 4.i.ii.iii: Finding the minimum element from the uncovered elements and substract it from the rest of the uncovered elements and adding it to the intersecting elements.

Step 4.iv. Finding the minimum number of lines required to cover all the zeroes.

Now we see that upon drawing the lines on Row 1, Row 2 and Column 2, all the zeroes are covered. But still the number of lines(m) is not equal to the number of rows/columns(n) and so we repeat step 4 again.

Step 4.i.ii.iii. Finding the minimum from the uncovered elements, substracting it from the rest of the uncovered elements and adding that element to the elements where lines are intersecting.

Now we see that on drawing vertical lines on row 1, row 2, row 3, row 4 we have covered all the zeroes in minimum number of lines. Now our number of lines(m)=number of rows/columns(n)=4 and so we have reached an optimal solution.

Step 5: Use the zeroes to assign possible combinations- wherever there is a zero present, task can be assigned to respective agent and job.

Step 6: Find the cost.

When we use the previous tables assignments of the jobs and cranes, we calculate the cost by adding the costs of the individual assignment. The minimum cost comes out to be 19 which tallies with the solution we got in the previous example. Hence we have arrived at the solution.

Note: If we assume the other combinations, we will still get the same answer of minimum value. The other combinations using the zeroes in the table obtained in 4(iv) gives us the same minimum cost.

Hope after reading this blog your concepts about the Assignment Problem and Hungarian Algorithm are clear!

Credits: The example used in this blog has been taken from the following YouTube video — https://www.youtube.com/watch?v=ezSx8OyBZVc . The problem statement is taken from — https://www.geeksforgeeks.org/hungarian-algorithm-assignment-problem-set-1-introduction/

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Implementing Hungarian Algorithm

Hello everybody,

A while ago I decided to learn about Hungarian algorithm. After watching some videos and reading some articles I think I got the main idea:

1) Find the minimum number in each row and subtract it from all elements in the row.

2) Find the minimum number in each column and subtract it from all elements in the column.

3) Cover all zeroes with minimum number of vertical and/or horizontal lines.

4) If the number of the lines is N, then we have found our assignment. Otherwise, find the minimum uncovered number, subtract it from all uncovered numbers and then go back to step 3.

Well, the thing is that I never found a good implementation. I can implement step 1 and 2. For step 3, I think that we can check whether the minimum number of lines is equal to N using bipartite matching. But if the minimum number is less than N, I don't know how I can find those covering lines. Please, help me with implementing it.

Thanks in advance!

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Hungarian Algorithm for Linear Assignment Problems (V2.3)

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Updated 15 Sep 2011

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This is an extremely fast implementation of the famous Hungarian algorithm (aslo known as Munkres' algorithm). It can solve a 1000 x 1000 problem in about 20 seconds in a Core Duo (T2500 @ 2.00GHz) XP laptop with Matlab 2008a, which is about 2.5 times faster than the mex code "assignmentoptimal" in FEX ID 6543, about 6 times faster than the author's first version in FEX ID 20328, and at least 30 times faster than other Matlab implementations in the FEX.

The code can also handle rectangular prolems and problems with forbiden allocations.

The new version (V2.3)is able to conduct a partial assignment if a full assignment is not feasible.

For more details of the Hungarian algorithm, visit http://csclab.murraystate.edu/bob.pilgrim/445/munkres.html

Yi Cao (2023). Hungarian Algorithm for Linear Assignment Problems (V2.3) (https://www.mathworks.com/matlabcentral/fileexchange/20652-hungarian-algorithm-for-linear-assignment-problems-v2-3), MATLAB Central File Exchange. Retrieved March 9, 2023 .

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Acknowledgements.

Inspired by: assignprob.zip , Functions for the rectangular assignment problem , Munkres Assignment Algorithm

Inspired: Hungarian Algorithm for Linear Sum Assignment Problem , Minimum Cost Constrained Input-Output and Control Configuration Co-Design Problem , Eigenshuffle , LAPJV - Jonker-Volgenant Algorithm for Linear Assignment Problem V3.0 , Hungarian based particle linking , simpletracker , Smooth Point-set Registration using Neighboring Constraints , TACTICS Toolbox

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Looking for a C/C++ implementation of the Hungarian method for real-valued cost matrix

I am looking for a C/C++ implementation of the Hungarian method for solving the linear assignment problem with real-valued cost matrix.

Some implementation I found, such as this one , only work for integer costs :(

Any suggestions are very appreciated!

Thank you in advance!

This one seems to work great for me:

https://github.com/mcximing/hungarian-algorithm-cpp

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Hungarian Method

The Hungarian method is a computational optimization technique that addresses the assignment problem in polynomial time and foreshadows following primal-dual alternatives. In 1955, Harold Kuhn used the term “Hungarian method” to honour two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry. Let’s go through the steps of the Hungarian method with the help of a solved example.

Hungarian Method to Solve Assignment Problems

The Hungarian method is a simple way to solve assignment problems. Let us first discuss the assignment problems before moving on to learning the Hungarian method.

What is an Assignment Problem?

A transportation problem is a type of assignment problem. The goal is to allocate an equal amount of resources to the same number of activities. As a result, the overall cost of allocation is minimised or the total profit is maximised.

Because available resources such as workers, machines, and other resources have varying degrees of efficiency for executing different activities, and hence the cost, profit, or loss of conducting such activities varies.

Assume we have ‘n’ jobs to do on ‘m’ machines (i.e., one job to one machine). Our goal is to assign jobs to machines for the least amount of money possible (or maximum profit). Based on the notion that each machine can accomplish each task, but at variable levels of efficiency.

Hungarian Method Steps

Check to see if the number of rows and columns are equal; if they are, the assignment problem is considered to be balanced. Then go to step 1. If it is not balanced, it should be balanced before the algorithm is applied.

Step 1 – In the given cost matrix, subtract the least cost element of each row from all the entries in that row. Make sure that each row has at least one zero.

Step 2 – In the resultant cost matrix produced in step 1, subtract the least cost element in each column from all the components in that column, ensuring that each column contains at least one zero.

Step 3 – Assign zeros

Step 4 – Perform the Optimal Test

Step 5 – Draw the least number of straight lines to cover all of the zeros as follows:

(a) Highlight the rows that aren’t assigned.

(b) Label the columns with zeros in marked rows (if they haven’t already been marked).

(d) Continue with (b) and (c) until no further marking is needed.

(f) Simply draw the lines through all rows and columns that are not marked. If the number of these lines equals the order of the matrix, then the solution is optimal; otherwise, it is not.

Step 6 – Find the lowest cost factor that is not covered by the straight lines. Subtract this least-cost component from all the uncovered elements and add it to all the elements that are at the intersection of these straight lines, but leave the rest of the elements alone.

Step 7 – Continue with steps 1 – 6 until you’ve found the highest suitable assignment.

Hungarian Method Example

Use the Hungarian method to solve the given assignment problem stated in the table. The entries in the matrix represent each man’s processing time in hours.

$\begin{array}{l}\begin{bmatrix} & I & II & III & IV & V \\1 & 20 & 15 & 18 & 20 & 25 \\2 & 18 & 20 & 12 & 14 & 15 \\3 & 21 & 23 & 25 & 27 & 25 \\4 & 17 & 18 & 21 & 23 & 20 \\5 & 18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} $

With 5 jobs and 5 men, the stated problem is balanced.

$\begin{array}{l}A = \begin{bmatrix}20 & 15 & 18 & 20 & 25 \\18 & 20 & 12 & 14 & 15 \\21 & 23 & 25 & 27 & 25 \\17 & 18 & 21 & 23 & 20 \\18 & 18 & 16 & 19 & 20 \\\end{bmatrix}\end{array} $

Subtract the lowest cost element in each row from all of the elements in the given cost matrix’s row. Make sure that each row has at least one zero.

$\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 5 & 10 \\6 & 8 & 0 & 2 & 3 \\0 & 2 & 4 & 6 & 4 \\0 & 1 & 4 & 6 & 3 \\2 & 2 & 0 & 3 & 4 \\\end{bmatrix}\end{array} $

Subtract the least cost element in each Column from all of the components in the given cost matrix’s Column. Check to see if each column has at least one zero.

$\begin{array}{l}A = \begin{bmatrix}5 & 0 & 3 & 3 & 7 \\6 & 8 & 0 & 0 & 0 \\0 & 2 & 4 & 4 & 1 \\0 & 1 & 4 & 4 & 0 \\2 & 2 & 0 & 1 & 1 \\\end{bmatrix}\end{array} $

When the zeros are assigned, we get the following:

The present assignment is optimal because each row and column contain precisely one encircled zero.

Where 1 to II, 2 to IV, 3 to I, 4 to V, and 5 to III are the best assignments.

Hence, z = 15 + 14 + 21 + 20 + 16 = 86 hours is the optimal time.

Practice Question on Hungarian Method

Use the Hungarian method to solve the following assignment problem shown in table. The matrix entries represent the time it takes for each job to be processed by each machine in hours.

$\begin{array}{l}\begin{bmatrix}J/M & I & II & III & IV & V \\1 & 9 & 22 & 58 & 11 & 19 \\2 & 43 & 78 & 72 & 50 & 63 \\3 & 41 & 28 & 91 & 37 & 45 \\4 & 74 & 42 & 27 & 49 & 39 \\5 & 36 & 11 & 57 & 22 & 25 \\\end{bmatrix}\end{array} $

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Frequently Asked Questions on Hungarian Method

What is hungarian method.

The Hungarian method is defined as a combinatorial optimization technique that solves the assignment problems in polynomial time and foreshadowed subsequent primal–dual approaches.

What are the steps involved in Hungarian method?

The following is a quick overview of the Hungarian method: Step 1: Subtract the row minima. Step 2: Subtract the column minimums. Step 3: Use a limited number of lines to cover all zeros. Step 4: Add some more zeros to the equation.

What is the purpose of the Hungarian method?

When workers are assigned to certain activities based on cost, the Hungarian method is beneficial for identifying minimum costs.

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Index Assignment problem Hungarian algorithm Solve online

The Hungarian algorithm: An example

We consider an example where four jobs (J1, J2, J3, and J4) need to be executed by four workers (W1, W2, W3, and W4), one job per worker. The matrix below shows the cost of assigning a certain worker to a certain job. The objective is to minimize the total cost of the assignment.

Below we will explain the Hungarian algorithm using this example. Note that a general description of the algorithm can be found here .

Step 1: Subtract row minima

We start with subtracting the row minimum from each row. The smallest element in the first row is, for example, 69. Therefore, we substract 69 from each element in the first row. The resulting matrix is:

Step 2: Subtract column minima

Step 3: Cover all zeros with a minimum number of lines

We will now determine the minimum number of lines (horizontal or vertical) that are required to cover all zeros in the matrix. All zeros can be covered using 3 lines:

Step 4: Create additional zeros

First, we find that the smallest uncovered number is 6. We subtract this number from all uncovered elements and add it to all elements that are covered twice. This results in the following matrix:

Now we return to Step 3.

Again, We determine the minimum number of lines required to cover all zeros in the matrix. Now there are 4 lines required:

Because the number of lines required (4) equals the size of the matrix ( n =4), an optimal assignment exists among the zeros in the matrix. Therefore, the algorithm stops.

The optimal assignment

The following zeros cover an optimal assignment:

This corresponds to the following optimal assignment in the original cost matrix:

Thus, worker 1 should perform job 3, worker 2 job 2, worker 3 job 1, and worker 4 should perform job 4. The total cost of this optimal assignment is to 69 + 37 + 11 + 23 = 140.

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hungarian-algorithm 0.1.11

pip install hungarian-algorithm Copy PIP instructions

Released: May 24, 2020

Python 3 implementation of the Hungarian Algorithm for the assignment problem.

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License: MIT License

Author: Ben Chaplin

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Hungarian algorithm.

The function find_matching takes 3 inputs:

*See examples below.

Example 1 (maximum-weighted matching)

Suppose you're choosing 11 starting positions for a soccer team.

The 11 players submit their top 3 position choices, and it is your job to create the optimal team.

The situation can be modeled with a weighted bipartite graph :

Define a weighted bipartite graph in the following manner:

thus defining a complete bipartite graph G = (L ∪ R, E) with:

Then pass the graph as an input:

You will get the output:

If you only need the total weight:

Example 2 (minimum-weighted matching)

Suppose you manage a group of drivers delivering packages to various locations.

You estimate the time of delivery for each driver to deliver each package, and it is your job to save the most time.

This time, we will model the situation with a matrix:

Then pass the graph as an input (this time, remember to change matching_type = 'min' to find a minimum-weighted matching):

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The Hungarian Method: The following algorithm applies the above theorem to a given n × n cost matrix to ﬁnd an optimal assignment. Step 1. Subtract the smallest entry in each row from all the entries of its row. Step 2. Subtract the smallest entry in each column from all the entries of its column. Step 3.
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This function is an implementation of the Hungarian algorithm (also known as the Kuhn-Munkres algorithm) which runs in O (N3) time. It solves the optimal assignment problem. Below is the implementation of the above approach: Python import dlib def minCost (arr): assignment = dlib.max_cost_assignment (arr)
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