- DOI: 10.1007/978-3-540-68279-0_2
- Corpus ID: 9426884
The Hungarian method for the assignment problem
- Published in 50 Years of Integer… 1 March 1955
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A note on hungarian method for solving assignment problem, the optimal assignment problem: an investigation into current solutions, new approaches and the doubly stochastic polytope, an effective algorithm to solve assignment problems : opportunity cost approach, optimal assignment problem on record linkage, an application of the hungarian algorithm to solve traveling salesman problem.
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A simulation of the faculty-assignment problem: an integer programming approach, computational studies of randomized multidimensional assignment problems, optimal solution of an assignment problem as a special case of transportation problem, assignment problems by, 17 references, a combinatorial algorithm, solution of the personnel classification problem with the method of optimal regions, the problem of classification of personnel, history of mathematical programming : a collection of personal reminiscences, on representatives of subsets, on a personnel assignment problem, on the hitchcock distribution problem, 1. a certain zero-sum two-person game equivalent to the optimal assignment problem, über graphen und ihre anwendung auf determinantentheorie und mengenlehre, related papers.
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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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Harold W. Kuhn
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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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The Hungarian Method for the Assignment Problem Introduction by
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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.
| 82 | 83 | 69 | 92 | |
| 77 | 37 | 49 | 92 | |
| 11 | 69 | 5 | 86 | |
| 8 | 9 | 98 | 23 |
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:
| 13 | 14 | 0 | 23 | (-69) | |
| 40 | 0 | 12 | 55 | (-37) | |
| 6 | 64 | 0 | 81 | (-5) | |
| 0 | 1 | 90 | 15 | (-8) |
Step 2: Subtract column minima
Similarly, we subtract the column minimum from each column, giving the following matrix:
| 13 | 14 | 0 | 8 | |
| 40 | 0 | 12 | 40 | |
| 6 | 64 | 0 | 66 | |
| 0 | 1 | 90 | 0 | |
| (-15) |
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:
| 13 | 14 | 0 | 8 | ||
| 40 | 0 | 12 | 40 | ||
| 6 | 64 | 0 | 66 | ||
| 0 | 1 | 90 | 0 | ||
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:
| 7 | 8 | 0 | 2 | |
| 40 | 0 | 18 | 40 | |
| 0 | 58 | 0 | 60 | |
| 0 | 1 | 96 | 0 |
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:
| 7 | 8 | 0 | 2 | ||
| 40 | 0 | 18 | 40 | ||
| 0 | 58 | 0 | 60 | ||
| 0 | 1 | 96 | 0 |
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.
Solve your own problem online
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Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
- For each row of the matrix, find the smallest element and subtract it from every element in its row.
- Do the same (as step 1) for all columns.
- Cover all zeros in the matrix using minimum number of horizontal and vertical lines.
- Test for Optimality: If the minimum number of covering lines is n, an optimal assignment is possible and we are finished. Else if lines are lesser than n, we haven’t found the optimal assignment, 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.
Try it before moving to see the solution
Explanation for above simple example:
An example that doesn’t lead to optimal value in first attempt: In the above example, the first check for optimality did give us solution. What if we the number covering lines is less than n.
Time complexity : O(n^3), where n is the number of workers and jobs. This is because the algorithm implements the Hungarian algorithm, which is known to have a time complexity of O(n^3).
Space complexity : O(n^2), where n is the number of workers and jobs. This is because the algorithm uses a 2D cost matrix of size n x n to store the costs of assigning each worker to a job, and additional arrays of size n to store the labels, matches, and auxiliary information needed for the algorithm.
In the next post, we will be discussing implementation of the above algorithm. The implementation requires more steps as we need to find minimum number of lines to cover all 0’s using a program. References: http://www.math.harvard.edu/archive/20_spring_05/handouts/assignment_overheads.pdf https://www.youtube.com/watch?v=dQDZNHwuuOY
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A note on Hungarian method for solving assignment problem. Jayanta Dutta Subhas Chandra Pal. Computer Science, Mathematics. 2015. TLDR. Hungarian method is modified to find out the optimal solution of an assignment problem which reduces the computational cost of the method. Expand.
A paper that introduces a computational method for solving the assignment problem, based on the work of Hungarian mathematicians Kanizsa and Egervary. The method uses duality and transfers to find the optimal assignment of persons to jobs with given scores.
The Hungarian algorithm is a method to solve the assignment problem, which is a combinatorial optimization problem of finding the best way to assign a set of items to a set of groups. The lecture notes explain the steps of the algorithm with examples and diagrams.
1. INTRODUCTION The Hungarian Method [ 11 is an algorithm for solving assignment problems that is based on the work of D. Konig and J. Egervgry. In one possible interpretation, an assignment problem asks for the best assignment of a set of persons to a set of jobs, where the feasible assignments are ranked by the total scores or ratings of the ...
The author recounts how he learned Hungarian's paper on the assignment problem and developed the Hungarian Method in 1953. He also discusses the applications, extensions and references of the method in graph theory and combinatorial optimization.
Hungarian method for assignment problem Step 1. Subtract the entries of each row by the row minimum. Step 2. Subtract the entries of each column by the column minimum. Step 3. Make an assignment to the zero entries in the resulting matrix. A = M 17 10 15 17 18 M 6 10 20 12 5 M 14 19 12 11 15 M 7 16 21 18 6 M −10
A chapter from a book on integer programming that describes the origin and application of the Hungarian method, a combinatorial algorithm for solving the assignment problem. The author, Harold W. Kuhn, shares his personal reminiscences and references related to the method.
The Hungarian method for the assignment problem ... View the article/chapter PDF and any associated supplements and figures for a period of 48 hours. Article/Chapter can not be printed. Article/Chapter can not be downloaded. Article/Chapter can not be redistributed.
It is shown that ideas latent in the work of two Hungarian mathematicians may be exploited to yield a new method of solving this problem. Bibliography 1 König, D. , " Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre.
The Hungarian method for the assignment problem. H. W. Kuhn, Search for more papers by this author. H. W. Kuhn, Search for more papers by this author. First published: 15 October 2004. ... View the article PDF and any associated supplements and figures for a period of 48 hours. Article can not be printed.
Learn how to formulate and solve assignment problems using the Hungarian method, a simpler and faster technique than the transportation method. See examples, special cases and applications of assignment problems, such as the travelling salesman problem.
The classical method for solving this problem is Kuhn's Hungarian method [Kuh55]. This method is of major theoretical interest and is still used widely. It maintains a price for each object and an (incomplete) assignment of persons and objects. At each iteration, the method chooses an unassigned person and computes a
Step 3. Draw lines through appropriate rows and columns so that all the zero entries of the cost matrix are covered and the minimum number of such lines is used. Step 4. Test for Optimality: (i) If the minimum number of covering lines is n, an optimal assignment of zeros is possible and we are finished.
The classical solution to the assignment problem is given by the Hungarian or Kuhn-Munkres algorithm, originally proposed by H. W. Kuhn in 1955 [3] and refined by J. Munkres in 1957 [5]. The Hungarian algorithm solves the assignment problem in O(n3) time, where n is the size of one partition of the bipartite graph. This and other
I then realized that Egerváry's paper gave a computationally trivial method for reducing the general assignment problem to a 0-1 problem. Thus, by putting the two ideas together, the Hungarian Method was born. I tested the algorithm by solving 12 by 12 problems with random 3-digit ratings by hand.
Learn how to solve an assignment problem using the Hungarian algorithm with a four-by-four cost matrix. Follow the steps of subtracting row and column minima, covering zeros with lines, and creating additional zeros to find the optimal assignment.
PDF | In this paper, we focus on the solution procedure for fully interval assignment problem (FIAP), Hungarian method is considered into account. In... | Find, read and cite all the research you ...
Learn how to use the Hungarian algorithm to solve the assignment problem, where you have to assign agents to tasks with minimum cost. The web page explains the algorithm steps, gives examples, and provides C++, Java, Python, C# and Javascript code.
Abstract and Figures. The Hungarian method is a well-known method for solving the assignment problem. This method was developed and published in 1955. It was named the Hungarian method because two ...
2] for the Hungarian method algorithm of solving the problem. 2.1 Data collection, analysis and conclusion . In this section, we shall consider a computational study and comparison of the new alternate method of assignment by [7] and the Hungarian method for solving University of Port Harcourt tender-job assignment problem.
Assignment Problem corresponds with the product distribution between demand points and supply points. Many algorithms were suggested to find the optimal result. The purpose of this study is to propose an appropriate model to explore the solution to the assignment problem. This paper focuses on Hungarian Method.