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Gabrovšek's Multiple Hungarian method for solving k-Assignment Problem.
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License: MIT License (MIT)
Author: Hoang-Nhat Tran (inspiros)
Tags k-assignment, k-partite-matching, linear-assignment, bi-partite-matching, hungarian
Requires: Python >=3.7
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Project description
Kap : $k$-assignment problem solver.
This project implements Boštjan Gabrovšek 's Multiple Hungarian Methods for solving the $k$-Assignment Problem (or $k$-Partite Graph Matching Problem ), described in this paper .
Problem Formulation
$k$-Assignment (a.k.a. $k$-Partite Graph Matching) Problem is the extension of Linear Assignment (Bipartite Graph Matching) Problem . It is also traditionally referred to as Multidimensional Assignment Problem .
Formally, we seek a $k$-assignment of a given $k$-partite weighted graph $G = (V, E, \omega)$ with the minimum weight:
This repository provides the implementation of 6 algorithms proposed by Gabrovšek for solving this problem, which is decomposed into small binary sub-problems and tackled with using the Hungarian procedure. While this means we can generalize for an arbitrary number $k$ and use different algorithms other than Hungarian, the methods might not be the most efficient for certain cases (e.g. $k = 3$ a.k.a. the 3-index Assignment Problem ).
For more technical details, please refer to the paper or contact the authors, not me 😂.
I implemented this code for testing an idea in another project. After that, I decided to publish the code so that someone facing a similar problem can use. Further maintenance or performance tuning might be limited, but all contribution is welcome.
Requirements
- Python 3.7+
- scipy or lap or lapjv or lapsolver or munkres (depends on the backend to be used for solving Linear Assignment Problem)
Installation
Simply run the following command:
This is currently a pure Python project, but we may add Cython/C++ extension in the future.
Quick Start
Solving linear assignment problem.
For convenience, we provide kap.linear_assignment as a wrapper around backend functions. The available backends are:
- scipy : scipy.optimize.linear_sum_assignment 's documentation
- lap : https://github.com/gatagat/lap
- lapjv : https://github.com/src-d/lapjv
- lapsolver : https://github.com/cheind/py-lapsolver
- munkres : https://github.com/bmc/munkres
Note that we currently do NOT support sparse matrix. For a benchmark, please head to https://github.com/berhane/LAP-solvers .
The interface is unified as kap.linear_assignment returns a namedtuple of matches , matching_costs of matched edges, and optionally a list of free (or unmatched) vertices if called with return_free=True .
Solving $k$-Assignment Problem
kap.k_assignment is the equivalence for $k$-Assignment Problem. There are a few things to note about its parameters:
- cost_matrices : Sequence of $n (n - 1) / 2$ 2D cost matrices of pairwise matching costs between $n$ partites (might not be able to be represented as a 3D np.ndarray since partites can have different number of vertices) ordered as in itertools.combination(n, 2) . For example, $[C_{01}, C_{02}, C_{03}, C_{12}, C_{13}, C_{23}]$.
- algo : This should be one of the six proposed algorithms, namely "Am", "Bm", "Cm", "Dm", "Em", "Fm" . $\text{C}_m$ is set to be the default as it usually performs as good as random approaches while having a deterministic behavior.
It's return type shares the same structure as that of kap.linear_assignment but with some small differences:
- matches : Each element is the list of indices of matched vertices. For example, [(0, 0), (1, 1), (2, 0)] indicates that node 0 of partite 0, node 1 of partite 1, and node 0 of partite 3 are matched together. Note that it is NOT necessary for a match to contain at least one vertex from each partite (incomplete graph).
- matching_costs : Each element is the sum of pairwise costs of all edges formed by the matched vertices.
The following code reproduce the example described in the paper.
These code blocks are extracted from examples .
MIT licensed. See LICENSE.txt .
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Multiple Hungarian Method for k -Assignment Problem
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(Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI-1000 Ljubljana, Slovenia)
(Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia)
(Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia)
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Multiple Hungarian Method for k-Assignment Problem
2020, Mathematics
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm can be considered as a greedy version of Bm, whereas Em and Fm are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm Am in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm Am) to 0.08% over optimum (algorithm Em). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm Am) to...
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Assignment problems arise in different situation where we have to find an optimal way to assign n-objects to mother objects in an injective fashion. The assignment problems are a well studied topic in combinatorial optimization. These problems find numerous application in production planning, telecommunication VLSI design, economic etc. The assignment problems is a special case of Transportation problem. Depending on the objective we want to optimize, we obtain the typical assignment problems. Assignment problem is an important subject discussed in real physical world we endeavor in this paper to introduce a new approach to assignment problem namely, matrix ones assignment method or MOA-method for solving wide range of problem. An example using matrix ones assignment methods and the existing Hungarian method have been solved and compared it graphically. Also some of the variations and some special cases in assignment problem and its applications have been discussed in the paper.
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Different situations give rise to the assignment problem, where we must discover an optimal way to assign 'n' objects to 'm' in an bijective function. We have, in this research, propose the possibility of solving exactly the Linear Assignment Problem with a method that would be more efficient than the Hungarian method of exact solution. This method is based on applying a series of pairwise interchanges of assignments to a starting heuristically generated feasible solution, wherein each pairwise interchange is guaranteed to improve the objective function value of the feasible solution.It seems that our algorithm finds the optimal solution which is the same as one found by the Hungarian method, but in much simpler. 7980 M. Khalid et al.
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50 Years of Integer Programming 1958-2008 pp 29–47 Cite as
The Hungarian Method for the Assignment Problem
- Harold W. Kuhn 9
- First Online: 01 January 2009
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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.
- Graph Theory
- Combinatorial Optimization
- Integer Program
- Assignment Problem
- National Bureau
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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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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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The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm ...
In the literature [4-6,10], this problem is also referred to as the multidimensional assignment problem (MAP). For the case k = 3 we can also trace the name 3-index assignment problem or 3-dimensional assignment problem in the literature. When k = 2, it is well-known that the Hungarian algorithm solves the 2-assignment problem to optimality ...
Five new heuristics are introduced, two of which arise as natural improvements of Algorithm Am, and incorporate randomization, which provide a good compromise between quality of solutions and computation time. The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm ...
Multiple Hungarian Method for k-Assignment Problem Boštjan Gabrovšek 1,2, Tina Novak 1, Janez Povh 1,3, Darja Rupnik Poklukar 1 and Janez Žerovnik 1,3,* 1 Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, SI-1000 Ljubljana, Slovenia;
Gabrovšek's Multiple Hungarian method for solving k-Assignment Problem. Project description kap: k -Assignment Problem Solver This project implements Boštjan Gabrovšek 's Multiple Hungarian Methods for solving the k -Assignment Problem (or k -Partite Graph Matching Problem ), described in this paper. Background Problem Formulation
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm,...
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004).
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. ... Multiple Hungarian Method for <i>k</i>-Assignment Problem Boštjan Gabrovšek, Tina Novak, Janez Povh, Darja Rupnik Poklukar and Janez Žerovnik; Affiliations ...
Downloadable! The k -assignment problem (or, the k -matching problem) on k -partite graphs is an NP-hard problem for k ≥ 3 . In this paper we introduce five new heuristics. Two algorithms, B m and C m , arise as natural improvements of Algorithm A m from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, D m , E m , and F m , incorporate ...
Abstract: The k -assignment problem (or, the k -matching problem) on k -partite graphs is an NP-hard problem for k ≥ 3 . In this paper we introduce five new heuristics. Two algorithms, B m and C m , arise as natural improvements of Algorithm A m from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004).
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004).
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.
Abstract The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k ≥ 3. In this paper we introduce five new heuristics. Two algorithms, B m and C m, arise as natural improvements of Algorithm A m from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004).
1 I've just started learning about the Hungarian Method. Everywhere I look, I see that the Hungarian Method gives an optimal assignment/solution to the assignment problem at hand, which I understand. However, what if there are multiple different assignments with the same (optimal) cost?
The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k ≥ 3. In this paper we introduce five new heuristics. Two algorithms, B$_m$ and C$_m$, arise as natural improvements of Algorithm A$_m$ from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004).
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.
This purpose can be served by assigning multiple jobs to a single machine. The present paper proposes a Modified Hungarian Method for solving unbalanced assignment problems which gives the optimal policy of assignment of jobs to machines. A stepwise algorithm of proposed method is presented and the developed algorithm is also coded in Java SE 11.
My Research and Language Selection Sign into My Research Create My Research Account English; Help and support. Support Center Find answers to questions about products, access, use, setup, and administration.; Contact Us Have a question, idea, or some feedback? We want to hear from you.
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.
This purpose can be served by assigning multiple jobs to a single machine. The present paper proposes a Modified Hungarian Method for solving unbalanced assignment problems which gives the optimal ...
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.
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 ...
Since the matrix of table 1 is a square matrix, the problem is balanced. Table 2 to table 5 present the steps required to determine the appropriate job assignment to the machine. 1. Subtracting the minimum element from all the elements in the respective rows, the new table will be: Table 2. Matrix table after step 1.