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Subhash Suri
UC Santa Barbara
Assignment and Matching
1. A matching is a pairing of nodes—
collection
of disjoint edges
.
Worker
Job
1
2
4
A
B
C
D
3
2. Bipartite graph. Two node classes,
workers and jobs.
3. An edge
(
i,j
)
means worker
i
can do job
j
.
4. If weighted, then
c
(
i,j
)
is the
proﬁciency
of
i
at job
j
. (In unweighted case,
c
(
i,j
) = 1
.)
5. Workers to jobs assignment for
maximizing
total proﬁciency
.
6. Each worker assigned to at most one job
and vice versa, so this is a matching.
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View Full Document Subhash Suri
UC Santa Barbara
Applications
1. [Rooming Problem.] Dorm room
assignment. Graph
G
with students as
nodes. Weight
c
ij
is
compatibility
of pair
(
i,j
)
.
2. [Airline Pilot Assignment.]
•
Airlines need to form teams of captain
and ﬁrst oﬃcer.
•
α
i
is eﬀectiveness of
i
as captain.
•
β
i
is eﬀectiveness of
i
as 1st oﬃcer.
•
Seniority Rule: captain more senior.
•
Make edge weight
c
ij
=
‰
α
i
+
β
j
if
i
more senior
α
j
+
β
j
otherwise
3. In these applications, the graph is
not
bipartite. We will only study the bipartite
case.
Subhash Suri
UC Santa Barbara
More Applications
4. [Stable Marriage.]
•
Men
{
A,B,.
..,Z
}
, women
{
a,b,.
..,z
}
.
•
Their preference tables.
A
B
C
Men’s Preferences
b
c
a
b
a
c
c
a
b
C
B
C
B
a
b
c
C
B
A
A
A
Women’s Prefereces
•
A matching
M
.
•
M
is unstable if
∃
pair
(
Bob,Sally
)
who like
each other more than their spouses.
•
Is stable marriage always possible?
•
Medical schools use this protocol.
•
GaleShapely Theorem: A stable marriage
always possible, and found in
O
(
n
2
)
time.
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View Full Document Subhash Suri
UC Santa Barbara
Stereo Vision
1. Stereo matching to locate objects in space.
2. Infrared sensors at two diﬀerent locations.
3. Each sensor gives the angle of sight (line)
on which the object lies.
.
.
.
.
.
.
.
.
.
.
.
.
O1
O2
l1
ln
h1
hn
4. If
p
objects, we get two sets of lines:
{
L
1
,L
2
,...,L
p
}
and
{
L
0
1
,L
0
2
,...,L
0
p
}
.
Subhash Suri
UC Santa Barbara
Stereo Vision
.
.
.
.
.
.
.
.
.
.
.
.
O1
O2
l1
ln
h1
hn
1. Two problems: (1) a line from one sensor
might intersect multiple lines from the
other; (2) due to noise, the lines for the
same object may not intersect.
2. Solve the problem using assingment.
Nodes are lines. Cost
c
ij
is the distance
between
L
i
and
L
0
j
.
3. Distance between lines of the same object
should be close to zero.
4. Optimal assingment should give excellent
matching of line.
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View Full Document
UC Santa Barbara
Deﬁnitions
1. A
matching
M
⊆
E
, in graph
G
= (
V,E
)
, is a
set of edges no two sharing a vertex.
A matching M
nonmatching edges
matching edges
2.

M

is the
cardinality
of
M
.
3. In unweighted graphs, ﬁnd max
cardinality matching.
4. In weighted graphs, ﬁnd max weight
matching.
5. A matching is perfect if all vertices are
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This note was uploaded on 12/27/2011 for the course CMPSC 225 taught by Professor Vandam during the Fall '09 term at UCSB.
 Fall '09
 Vandam

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