CS 7250, Approximation Algorithms
Homework 3
Thu, Feb 17, 2011
Due Thu, Feb 24, 2011
Problem 1: PrimalDual, Exact Complementary Slackness
This exercise is a review of the basics of LPduality (chaper 12 in Vazirani’s book).
(a)
Let
G
(
V, E
) be an undirected graph with
source s
∈
V
and
sink t
∈
V
\{
s
}
, and edge capacities
c
:
E
→
R
+
. Replace each undirected edge
{
i, j
}
with two directed edges (
i, j
) and (
j, i
) of equal
capacities
c
ij
=
c
ji
. Introduce a link of infinite capacity from
t
to
s
, and a link of zero capacity
from
s
to
t
:
c
ts
=
∞
and
c
st
=0. Then the straightforward formalization of max
s

t
flow in
G
is:
maximize
f
ts
subject to
f
ij
≤
c
ij
,
∀
(
i, j
)
∈
E
∑
j
:(
j,i
)
∈
E
f
ji

∑
j
:(
i,j
)
∈
E
f
ij
= 0
,
∀
i
∈
V
(1)
f
ij
≥
0
,
∀
(
i, j
)
∈
E
In class we formalized max
s

t
flow with the slightly different LP (2) below, where the set of
flow preservation constraints of (1) are replaced with a set of corresponding constraints that may
initially seem weaker.
maximize
f
ts
subject to
f
ij
≤
c
ij
,
∀
(
i, j
)
∈
E
∑
j
:(
j,i
)
∈
E
f
ji

∑
j
:(
i,j
)
∈
E
f
ij
≤
0
,
∀
i
∈
V
(2)
f
ij
≥
0
,
∀
(
i, j
)
∈
E
Prove that the constraints in (2):
∑
j
:(
j,i
)
∈
E
f
ji

∑
j
:(
i,j
)
∈
E
f
ij
≤
0
,
∀
i
∈
V
, imply the flow preser
vation constraints in (1):
∑
j
:(
j,i
)
∈
E
f
ji

∑
j
:(
i,j
)
∈
E
f
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 Spring '08
 Staff
 Algorithms, vertex cover, Approximation algorithm, complementary slackness, Vazirani, max st ﬂow

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