Com S 511: Homework #3
Due on Friday, October 2, 2009
Yuheng Long
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentYuheng Long
Com S 511 : Homework #3
Problem 1
Problem 1
Problem 2
We will have the following constraints:
•
The aggregate ﬂow on an edge e must be no more than the capacity of the edge, c(e).
f
(
u,v
) =
∑
k
i
=1
f
i
(
u,v
)
≤
c
(
u,v
) for each
u,v
∈
V
•
For an internal node, the amount of ﬂow entering must equal the amount of ﬂow leaving.
∑
v
∈
V
f
i
(
u,v
) = 0 for each
u
∈
V
 {
s
i
,t
i
}
•
For a source(sink)
s
i
(
t
i
) node, the amount of ﬂow leaving it must be exactly
d
i
larger(smaller) than
the amount of ﬂow entering.
∑
v
∈
V
f
i
(
s
i
,v
) =
∑
v
∈
V
f
i
(
v,t
i
) =
d
i
We want to minimize the cost. Therefore,
minimize
∑
u,v
∈
E
(
a
(
u,v
)
∑
k
i
=1
f
i
(
u,v
))
subject to
∑
k
i
=1
f
i
(
u,v
)
≤
c
(
u,v
)
for each
u,v
∈
V
∑
v
∈
V
f
i
(
v,u
) =
∑
v
∈
V
f
i
(
u,v
)
for each
i
= 1
,
2
,...,k
and for each
u
∈
V
{
s
i
,t
i
}
∑
v
∈
V
f
i
(
s
i
,v
) =
∑
v
∈
V
f
i
(
v,t
i
) =
d
i
for each
i
= 1
,
2
,...,k
f
i
(
u,v
)
≥
0
for each
i
= 1
,
2
,...,k
and for each
u,v
∈
V
Problem 3
Consider the following linear program:
maximize
x

y
subject to
ax
+
by
≤ 
1
x,y
≥
0
,
(1)
where
a
and
b
are real numbers.
(a) The LP is infeasible if and only if
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Optimization, LP

Click to edit the document details