assignment problem analysis of algorithm

The assignment problem revisited

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• Published: 16 August 2021
• Volume 16 , pages 1531–1548, ( 2022 )

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• Carlos A. Alfaro   ORCID: orcid.org/0000-0001-9783-8587 1 ,
• Sergio L. Perez 2 ,
• Carlos E. Valencia 3 &
• Marcos C. Vargas 1  

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First, we give a detailed review of two algorithms that solve the minimization case of the assignment problem, the Bertsekas auction algorithm and the Goldberg & Kennedy algorithm. It was previously alluded that both algorithms are equivalent. We give a detailed proof that these algorithms are equivalent. Also, we perform experimental results comparing the performance of three algorithms for the assignment problem: the \(\epsilon \) - scaling auction algorithm , the Hungarian algorithm and the FlowAssign algorithm . The experiment shows that the auction algorithm still performs and scales better in practice than the other algorithms which are harder to implement and have better theoretical time complexity.

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Some results on an assignment problem variant

Integer Programming

A Full Description of Polytopes Related to the Index of the Lowest Nonzero Row of an Assignment Matrix

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This research was partially supported by SNI and CONACyT.

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Alfaro, C.A., Perez, S.L., Valencia, C.E. et al. The assignment problem revisited. Optim Lett 16 , 1531–1548 (2022). https://doi.org/10.1007/s11590-021-01791-4

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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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Quadratic assignment problem

Author: Thomas Kueny, Eric Miller, Natasha Rice, Joseph Szczerba, David Wittmann (SysEn 5800 Fall 2020)

• 1 Introduction
• 2.1 Koopmans-Beckman Mathematical Formulation
• 2.2.1 Parameters
• 2.3.1 Optimization Problem
• 2.4 Computational Complexity
• 2.5 Algorithmic Discussions
• 2.6 Branch and Bound Procedures
• 2.7 Linearizations
• 3.1 QAP with 3 Facilities
• 4.1 Inter-plant Transportation Problem
• 4.2 The Backboard Wiring Problem
• 4.3 Hospital Layout
• 4.4 Exam Scheduling System
• 5 Conclusion
• 6 References

Introduction

The Quadratic Assignment Problem (QAP), discovered by Koopmans and Beckmann in 1957 [1] , is a mathematical optimization module created to describe the location of invisible economic activities. An NP-Complete problem, this model can be applied to many other optimization problems outside of the field of economics. It has been used to optimize backboards, inter-plant transportation, hospital transportation, exam scheduling, along with many other applications not described within this page.

Theory, Methodology, and/or Algorithmic Discussions

Koopmans-beckman mathematical formulation.

Economists Koopmans and Beckman began their investigation of the QAP to ascertain the optimal method of locating important economic resources in a given area. The Koopmans-Beckman formulation of the QAP aims to achieve the objective of assigning facilities to locations in order to minimize the overall cost. Below is the Koopmans-Beckman formulation of the QAP as described by neos-guide.org.

Quadratic Assignment Problem Formulation

Inner Product

Note: The true objective cost function only requires summing entries above the diagonal in the matrix comprised of elements

Since this matrix is symmetric with zeroes on the diagonal, dividing by 2 removes the double count of each element to give the correct cost value. See the Numerical Example section for an example of this note.

Optimization Problem

With all of this information, the QAP can be summarized as:

Computational Complexity

QAP belongs to the classification of problems known as NP-complete, thus being a computationally complex problem. QAP’s NP-completeness was proven by Sahni and Gonzalez in 1976, who states that of all combinatorial optimization problems, QAP is the “hardest of the hard”. [2]

Algorithmic Discussions

While an algorithm that can solve QAP in polynomial time is unlikely to exist, there are three primary methods for acquiring the optimal solution to a QAP problem:

• Dynamic Program
• Cutting Plane

Branch and Bound Procedures

The third method has been proven to be the most effective in solving QAP, although when n > 15, QAP begins to become virtually unsolvable.

The Branch and Bound method was first proposed by Ailsa Land and Alison Doig in 1960 and is the most commonly used tool for solving NP-hard optimization problems.

A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores branches of this tree, which represent subsets of the solution set. Before one lists all of the candidate solutions of a branch, the branch is checked against upper and lower estimated bounds on the optimal solution, and the branch is eliminated if it cannot produce a better solution than the best one found so far by the algorithm.

Linearizations

The first attempts to solve the QAP eliminated the quadratic term in the objective function of

in order to transform the problem into a (mixed) 0-1 linear program. The objective function is usually linearized by introducing new variables and new linear (and binary) constraints. Then existing methods for (mixed) linear integer programming (MILP) can be applied. The very large number of new variables and constraints, however, usually poses an obstacle for efficiently solving the resulting linear integer programs. MILP formulations provide LP relaxations of the problem which can be used to compute lower bounds.

Numerical Example

Qap with 3 facilities.

Applications

Inter-plant transportation problem.

The QAP was first introduced by Koopmans and Beckmann to address how economic decisions could be made to optimize the transportation costs of goods between both manufacturing plants and locations. [1] Factoring in the location of each of the manufacturing plants as well as the volume of goods between locations to maximize revenue is what distinguishes this from other linear programming assignment problems like the Knapsack Problem.

The Backboard Wiring Problem

As the QAP is focused on minimizing the cost of traveling from one location to another, it is an ideal approach to determining the placement of components in many modern electronics. Leon Steinberg proposed a QAP solution to optimize the layout of elements on a blackboard by minimizing the total amount of wiring required. [4]

When defining the problem Steinberg states that we have a set of n elements

as well as a set of r points

In his paper he derives the below formula:

In his paper Steinberg a backboard with a 9 by 4 array, allowing for 36 potential positions for the 34 components that needed to be placed on the backboard. For the calculation, he selected a random initial placement of s1 and chose a random family of 25 unconnected sets.

The initial placement of components is shown below:

After the initial placement of elements, it took an additional 35 iterations to get us to our final optimized backboard layout. Leading to a total of 59 iterations and a final wire length of 4,969.440.

Hospital Layout

Building new hospitals was a common event in 1977 when Alealid N Elshafei wrote his paper on "Hospital Layouts as a Quadratic Assignment Problem". [5] With the high initial cost to construct the hospital and to staff it, it is important to ensure that it is operating as efficiently as possible. Elshafei's paper was commissioned to create an optimization formula to locate clinics within a building in such a way that minimizes the total distance that a patient travels within the hospital throughout the year. When doing a study of a major hospital in Cairo he determined that the Outpatient ward was acting as a bottleneck in the hospital and focused his efforts on optimizing the 17 departments there.

Elshafei identified the following QAP to determine where clinics should be placed:

For the Cairo hospital with 17 clinics, and one receiving and recording room bringing us to a total of 18 facilities. By running the above optimization Elshafei was able to get the total distance per year down to 11,281,887 from a distance of 13,973,298 based on the original hospital layout.

Exam Scheduling System

The scheduling system uses matrices for Exams, Time Slots, and Rooms with the goal of reducing the rate of schedule conflicts. To accomplish this goal, the “examination with the highest cross faculty student is been prioritized in the schedule after which the examination with the highest number of cross-program is considered and finally with the highest number of repeating student, at each stage group with the highest number of student are prioritized.” [6]

• ↑ 1.0 1.1 1.2 Koopmans, T., & Beckmann, M. (1957). Assignment Problems and the Location of Economic Activities. Econometrica, 25(1), 53-76. doi:10.2307/1907742
• ↑ 2.0 2.1 Quadratic Assignment Problem. (2020). Retrieved December 14, 2020, from https://neos-guide.org/content/quadratic-assignment-problem
• ↑ 3.0 3.1 3.2 Burkard, R. E., Çela, E., Pardalos, P. M., & Pitsoulis, L. S. (2013). The Quadratic Assignment Problem. https://www.opt.math.tugraz.at/~cela/papers/qap_bericht.pdf .
• ↑ 4.0 4.1 Leon Steinberg. The Backboard Wiring Problem: A Placement Algorithm. SIAM Review . 1961;3(1):37.
• ↑ 5.0 5.1 Alwalid N. Elshafei. Hospital Layout as a Quadratic Assignment Problem. Operational Research Quarterly (1970-1977) . 1977;28(1):167. doi:10.2307/300878
• ↑ 6.0 6.1 Muktar, D., & Ahmad, Z.M. (2014). Examination Scheduling System Based On Quadratic Assignment.

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Hungarian Algorithm for Assignment Problem | Set 2 (Implementation)

• Hungarian Algorithm for Assignment Problem | Set 1 (Introduction)
• Implementation of Exhaustive Search Algorithm for Set Packing
• Greedy Approximate Algorithm for Set Cover Problem
• Introduction to Exact Cover Problem and Algorithm X
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• Java Program for Counting sets of 1s and 0s in a binary matrix
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• C / C++ Program for Dijkstra's shortest path algorithm | Greedy Algo-7
• Self assignment check in assignment operator
• Python Program for Dijkstra's shortest path algorithm | Greedy Algo-7
• Algorithms | Dynamic Programming | Question 7
• Assignment Operators in C
• Assignment Operators in Programming

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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