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Assignment Problem:
Hungarian Algorithm and Linear Programming
collected from the Internet and extended by
Longin Jan Latecki
Slide 1 of 27
Assignment Problem Exampl




An assignment problem seeks to minimize the total cost
assignment of m workers to m jobs, given that the cost
of worker i performing job j is cij.
It assumes all workers are assigned and each job is
performed.
An assignment problem is a special case of a
transportation problem in which all supplies and all
demands are equal to 1; hence assignment problems
may be solved as linear programs.
The graph representation of an assignment problem
with three workers and three jobs is shown on the next
slide.
Slide 2 of 27
Assignment Problem

Graph (or Network) Representation
1
c11
c12
1
c13
c21
2
c22
2
c23
c32
c31
3
3
c33
WORKERS
JOBS
Slide 3 of 27
Introduction
to Assignment Problem
Let
C be an nxn matrix representing the costs of each of n workers to perform any of n
jobs. The assignment problem is to assign jobs to workers so as to minimize the total
cost. Since each worker can perform only one job and each job can be assigned to only
one worker the assignments constitute an independent set of the matrix C.
An
arbitrary assignment is shown above in which worker a is assigned job q, worker b
is assigned job s and so on. The total cost of this assignment is 23.
Can you find a lower cost assignment?
Can you find the minimal cost assignment?
Remember that each assignment must be unique in its row and column.
Slide 4 of 27
A
brute-force algorithm for solving the assignment problem involves generating all
independent sets of the matrix C, computing the total costs of each assignment and a
search of all assignment to find a minimal-sum independent set. The complexity of this
method is driven by the number of independent assignments possible in an nxn matrix.
There are n choices for the first assignment, n-1 choices for the second assignment and
so on, giving n! possible assignment sets. Therefore, this approach has, at least, an
factorial runtime complexity.
As
each assignment is chosen that row and column are eliminated from
consideration. The question is raised as to whether there is a better algorithm. In fact
there exists a polynomial runtime complexity algorithm.
Slide 5 of 27
Example 1: AP
A contractor pays his subcontractors a fixed fee plus mileage for
work performed. On a given day the contractor is faced with three
electrical jobs associated with various projects. Given below are
the distances between the subcontractors and the projects.
Project
A B C
Westside
50 36 16
Subcontractors
Federated
28 30 18
Goliath
35 32 20
Universal
25 25 14
How should the contractors be assigned to minimize total costs?
Note: There are four subcontractors and three projects. We create a
dummy project Dum, which will be assigned to one subcontractor
(i.e. that subcontractor will remain idle)
Slide 6 of 27
Example 1: AP

Initial Tableau Setup
Since the Hungarian algorithm requires that there
be the same number of rows as columns, add a Dummy
column so that the first tableau is (the smallest elements
in each row are marked red):
Westside
Federated
Goliath
Universal
A
50
28
35
25
B
36
30
32
25
C Dummy
16
0
18
0
20
0
14
0
Slide 7 of 27
Example 1: AP

Network Representation (note the dummy project)
50
West.
A
36
16
0
28
Fed.
30
18
B
0
32
35
20
Gol.
C
0
25
25
Univ.
14
0
Dum.
Slide 8 of 27
Assignment Problem

Linear Programming Formulation
Min SScijxij
ij
s.t. Sxij = 1
j
for each resource (row) i
Sxij = 1
for each job (column) j
i
xij = 0 or 1 for all i and j.
• Note: A modification to the right-hand side of the first
constraint set can be made if a worker is permitted to work
more than 1 job.
Slide 9 of 27
Example 1: AP via LP

In our example the LP formulation is:
Min z = 50x11 + 36x12 + 16x13 + 0x14 + 28x21 + 30x22 + 18x23 + 0x24 +
35x31 + 32x32 + 20x33 + 0x34 + + 25x41 + 25x42 + 14x43 + 0x44
s.t.
x11 + x12 + x13 + x14 = 1
(row 1)
x21 + x22 + x23 + x24 = 1
(row 2)
x31 + x32 + x33 + x34 = 1
(row 3)
x41 + x42 + x43 + x44 = 1
(row 4)
x11 + x21 + x31 + x41 = 1
(column 1)
x12 + x22 + x32 + x42 = 1
(column 2)
x13 + x23 + x33 + x43 = 1
(column 3)
x14 + x24 + x34 + x44 = 1
(column 4)
xij >= 0 for i = 1, 2, 3, 4 and j = 1, 2, 3, 4 (nonnegativity)
Slide 10 of 27
Example 1: AP via LP

The solver formulation is:
The Assignment Problem
Contractor
West Side
Federated
Goliath
Universal
Distances From
Contractors to Projects at:
A
B
C
50
36
16
28
30
18
35
32
20
25
25
14
Dummy
0
0
0
0
Contractor
West Side
Federated
Goliath
Universal
Assigned
Capacity
Assignment of Contractors
to Projects at:
A
B
C
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
Dummy
0
0
0
0
0
1
Total Distance:
Assigned Available
0
1
0
1
0
1
0
1
0
Slide 11 of 27
Example 1: AP via LP

The solver solution is:
The Assignment Problem
Contractor
West Side
Federated
Goliath
Universal
Distances From
Contractors to Projects at:
A
B
C
50
36
16
28
30
18
35
32
20
25
25
14
Dummy
0
0
0
0
Contractor
West Side
Federated
Goliath
Universal
Assigned
Capacity
Assignment of Contractors
to Projects at:
A
B
C
0
0
1
1
0
0
0
0
0
0
1
0
1
1
1
1
1
1
Dummy
0
0
1
0
1
1
Total Distance:
Assigned Available
1
1
1
1
1
1
1
1
69
Slide 12 of 27

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Linear Programming (LP) problems can be solved on
the computer using Matlab, and many others.
There are special classes of LP problems such as the
assignment problem (AP).
Efficient solutions methods exist to solve AP.
AP can be formulated as an LP and solved by general
purpose LP codes.
However, there are many computer packages, which
contain separate computer codes for these models
which take advantage of the problem network
structure.
Slide 13 of 27
Hungarian Method


The Hungarian method solves minimization
assignment problems with m workers and m jobs.
Special considerations can include:
• number of workers does not equal the number of
jobs — add dummy workers/jobs with 0 assignment
costs as needed
• worker i cannot do job j — assign cij = +M
• maximization objective — create an opportunity loss
matrix subtracting all profits for each job from the
maximum profit for that job before beginning the
Hungarian method
Slide 14 of 27
From
en.wikipedia.org/wiki/Hungarian_algorithm
The Hungarian method is a combinatorial optimization
algorithm which solves the assignment problem in polynomial
time. It was developed and published by Harold Kuhn in 1955,
who gave the name "Hungarian method" because the
algorithm was largely based on the earlier works of two
Hungarian mathematicians: Dénes Kőnig and Jenő Egerváry.
James Munkres reviewed the algorithm in 1957 and observed
that it is (strongly) polynomial. Since then the algorithm has
been known also as Kuhn-Munkres or Munkres assignment
algorithm. The time complexity of the original algorithm was
O(n4), however later it was noticed that it can be modified to
achieve an O(n3) running time.
In 2006, it was discovered that Carl Gustav Jacobi had solved
the assignment problem in the 19th century, and published
posthumously in 1890 in Latin.
Slide 15 of 27
Hungarian Method
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Step 1: For each row, subtract the minimum number in
that row from all numbers in that row.
Step 2: For each column, subtract the minimum number
in that column from all numbers in that column.
Step 3: Draw the minimum number of lines to cover all
zeroes. If this number = m, STOP — an assignment can
be made.
Step 4: Determine the minimum uncovered number
(call it d).
• Subtract d from uncovered numbers.
• Add d to numbers covered by two lines.
• Numbers covered by one line remain the same.
• Then, GO TO STEP 3.
Slide 16 of 27
Hungarian Method

Finding the Minimum Number of Lines and
Determining the Optimal Solution
• Step 1: Find a row or column with only one unlined
zero and circle it. (If all rows/columns have two or
more unlined zeroes choose an arbitrary zero.)
• Step 2: If the circle is in a row with one zero, draw a
line through its column. If the circle is in a column
with one zero, draw a line through its row. One
approach, when all rows and columns have two or
more zeroes, is to draw a line through one with the
most zeroes, breaking ties arbitrarily.
• Step 3: Repeat step 2 until all circles are lined. If this
minimum number of lines equals m, the circles
provide the optimal assignment.
Slide 17 of 27
Example 1: AP

Step 1: Subtract minimum number in each row from all
numbers in that row. Since each row has a zero, we
simply generate the original matrix (the smallest
elements in each column are marked red). This yields:
Westside
Federated
Goliath
Universal
A B C
50 36 16
28 30 18
35 32 20
25 25 14
Dummy
0
0
0
0
Slide 18 of 27
Example 1: AP

Step 2: Subtract the minimum number in each column
from all numbers in the column. For A it is 25, for B it
is 25, for C it is 14, for Dummy it is 0. This yields:
Westside
Federated
Goliath
Universal
A
25
3
10
0
B
11
5
7
0
C Dummy
2
0
4
0
6
0
0
0
Slide 19 of 27
Example 1: AP

Step 3: Draw the minimum number of lines to cover all zeroes
(called minimum cover). Although one can "eyeball" this
minimum, use the following algorithm. If a "remaining" row has
only one zero, draw a line through the column. If a remaining
column has only one zero in it, draw a line through the row. Since
the number of lines that cover all zeros is 2 < 4 (# of rows), the
current solution is not optimal.
Westside
Federated
Goliath
Universal

A
25
3
10
0
B
11
5
7
0
C
2
4
6
0
Dummy
0
0
0
0
Step 4: The minimum uncovered number is 2 (circled).
Slide 20 of 27
Example 1: AP

Step 5: Subtract 2 from uncovered numbers; add 2 to all
numbers at line intersections; leave all other numbers
intact. This gives:
Westside
Federated
Goliath
Universal
A
23
1
8
0
B
9
3
5
0
C Dummy
0
0
2
0
4
0
0
2
Slide 21 of 27
Example 1: AP

Step 3: Draw the minimum number of lines to cover all
zeroes. Since 3 (# of lines) < 4 (# of rows), the current
solution is not optimal.
A
B
C Dummy
Westside
23
9
0
0
Federated
1
3
2
0
Goliath
8
5
4
0
Universal
0
0
0
2

Step 4: The minimum uncovered number is 1 (circled).
Slide 22 of 27
Example 1: AP

Step 5: Subtract 1 from uncovered numbers. Add 1 to
numbers at intersections. Leave other numbers intact.
This gives:
Westside
Federated
Goliath
Universal
A
23
0
7
0
B
9
2
4
0
C
0
1
3
0
Dummy
1
0
0
3
Slide 23 of 27
Example 1: AP
Find the minimum cover:
Westside
Federated
Goliath
Universal

A
23
0
7
0
B
9
2
4
0
C
0
1
3
0
Dummy
1
0
0
3
Step 4: The minimum number of lines to cover all 0's is
four. Thus, the current solution is optimal (minimum
cost) assignment.
Slide 24 of 27
Example 1: AP
The optimal assignment occurs at locations of zeros
such that there is exactly one zero in each row and each
column:
Westside
Federated
Goliath
Universal
A
23
0
7
0
B
9
2
4
0
C
0
1
3
0
Dummy
1
0
0
3
Slide 25 of 27
Example 1: AP
The optimal assignment is (go back to the original table
for the distances):
Subcontractor Project Distance
Westside
C
16
Federated
A
28
Universal
B
25
Goliath
(unassigned)
Total Distance = 69 miles
Slide 26 of 27