assignment algorithm hungarian method

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

• Analyse the rows one by one until you find a row with precisely one unmarked zero. Encircle this lonely unmarked zero and assign it a task. All other zeros in the column of this circular zero should be crossed out because they will not be used in any future assignments. Continue in this manner until you’ve gone through all of the rows.
• Examine the columns one by one until you find one with precisely one unmarked zero. Encircle this single unmarked zero and cross any other zero in its row to make an assignment to it. Continue until you’ve gone through all of the columns.

Step 4 – Perform the Optimal Test

• The present assignment is optimal if each row and column has exactly one encircled zero.
• The present assignment is not optimal if at least one row or column is missing an assignment (i.e., if at least one row or column is missing one encircled zero). Continue to step 5. Subtract the least cost element from all the entries in each column of the final cost matrix created in step 1 and ensure that each column has at least one zero.

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).

(c) Highlight the rows that have assignments in indicated columns (if they haven’t previously 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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Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)

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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:

• For each row of the matrix, find the smallest element and subtract it from every element in its row.
• Repeat the step 1 for all columns.
• Cover all zeros in the matrix using the minimum number of horizontal and vertical lines.
• Test for Optimality : If the minimum number of covering lines is N , an optimal assignment is possible. Else if lines are lesser than N , an optimal assignment is not found and must proceed to step 5.
• Determine the smallest entry not covered by any line. Subtract this entry from each uncovered row, and then add it to each covered column. Return to step 3.

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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Hungarian Method Examples

Now we will examine a few highly simplified illustrations of Hungarian Method for solving an assignment problem .

Later in the chapter, you will find more practical versions of assignment models like Crew assignment problem , Travelling salesman problem , etc.

Example-1, Example-2

Example 1: Hungarian Method

The Funny Toys Company has four men available for work on four separate jobs. Only one man can work on any one job. The cost of assigning each man to each job is given in the following table. The objective is to assign men to jobs in such a way that the total cost of assignment is minimum.

This is a minimization example of assignment problem . We will use the Hungarian Algorithm to solve this problem.

Identify the minimum element in each row and subtract it from every element of that row. The result is shown in the following table.

"A man has one hundred dollars and you leave him with two dollars, that's subtraction." -Mae West

On small screens, scroll horizontally to view full calculation

Identify the minimum element in each column and subtract it from every element of that column.

Make the assignments for the reduced matrix obtained from steps 1 and 2 in the following way:

• For every zero that becomes assigned, cross out (X) all other zeros in the same row and the same column.
• If for a row and a column, there are two or more zeros and one cannot be chosen by inspection, choose the cell arbitrarily for assignment.

An optimal assignment is found, if the number of assigned cells equals the number of rows (and columns). In case you have chosen a zero cell arbitrarily, there may be alternate optimal solutions. If no optimal solution is found, go to step 5.

Use Horizontal Scrollbar to View Full Table Calculation

Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix obtained from step 3 by adopting the following procedure:

• Mark all the rows that do not have assignments.
• Mark all the columns (not already marked) which have zeros in the marked rows.
• Mark all the rows (not already marked) that have assignments in marked columns.
• Repeat steps 5 (ii) and (iii) until no more rows or columns can be marked.
• Draw straight lines through all unmarked rows and marked columns.

You can also draw the minimum number of lines by inspection.

Select the smallest element (i.e., 1) from all the uncovered elements. Subtract this smallest element from all the uncovered elements and add it to the elements, which lie at the intersection of two lines. Thus, we obtain another reduced matrix for fresh assignment.

Now again make the assignments for the reduced matrix.

Final Table: Hungarian Method

Since the number of assignments is equal to the number of rows (& columns), this is the optimal solution.

The total cost of assignment = A1 + B4 + C2 + D3

Substituting values from original table: 20 + 17 + 17 + 24 = Rs. 78.

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Hungarian Algorithm Introduction & Python Implementation

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

By Eason on 2021-08-02

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:

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:

• Mark rows that do not contain marked 0 elements and store row indexes in the non_marked_row
• Search non_marked_row element, and find out if there are any unmarked 0 elements in the corresponding column
• Store the column indexes in the marked_cols
• Compare the column indexes stored in marked_zero and marked_cols
• If a matching column index exists, the corresponding row_index is saved to non_marked_rows
• Next, the row indexes that are not in non_marked_row are stored in marked_rows

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:

• marked_zero : [(3, 2), (0, 4), (1, 1), (2, 0), (4, 3)]
• marked_rows : [0, 1, 2, 3, 4]
• marked_cols : []

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 Hungarian Method for the Assignment Problem

• First Online: 01 January 2009

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• Harold W. Kuhn 9  

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This paper has always been one of my favorite “children,” combining as it does elements of the duality of linear programming and combinatorial tools from graph theory. It may be of some interest to tell the story of its origin.

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H.W. Kuhn, On the origin of the Hungarian Method , History of mathematical programming; a collection of personal reminiscences (J.K. Lenstra, A.H.G. Rinnooy Kan, and A. Schrijver, eds.), North Holland, Amsterdam, 1991, pp. 77–81.

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A. Schrijver, Combinatorial optimization: polyhedra and efficiency , Vol. A. Paths, Flows, Matchings, Springer, Berlin, 2003.

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Kuhn, H.W. (2010). The Hungarian Method for the Assignment Problem. In: Jünger, M., et al. 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68279-0_2

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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

Similarly, we subtract the column minimum from each column, giving the following matrix:

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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