Yuheng 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