be routed either clockwise or counterclockwise around the cycle from its source node
to its destination node. Our goal is to route the calls so as to minimize the overall
load on the network. The load
L
i
on any edge
(
i
,
(
i
+
1
)
mod
n
)
is the number of
calls routed through that edge, and the overall load is
max
i
L
i
. Describe and analyze
an efficient
approximation algorithm for this problem.
.
The
linear arrangement
problem asks, given an
n
vertex directed graph as input,
for an ordering
v
1
,
v
2
,...,
v
n
of the vertices that maximizes the number of forward
edges: directed edges
v
i
v
j
such that
i
<
j
. Describe and analyze an efficient

approximation algorithm for this problem. (Solving this problem exactly is NPhard.)
“
J.
. An FPTAS for Subset Sum
.
A
threedimensional matching
in an undirected graph
G
is a collection of vertex
disjoint triangles (cycles of length
3
) in
G
. A threedimensional matching is
maximal
if it is not a proper subgraph of a larger threedimensional matching in the same
graph.
(a)
Let
M
and
M
0
be two arbitrary maximal threedimensional matchings in the
same underlying graph
G
. Prove that

M
 ≤
3
· 
M
0

.
(b)
Finding the
largest
threedimensional matching in a given graph is NPhard.
Describe and analyze a fast
approximation algorithm for this problem.
(c)
Finding the
smallest maximal
threedimensional matching in a given graph is
NPhard. Describe and analyze a fast
approximation algorithm for this problem.
.
Consider the following optimization version of the
P
problem. Given a set
X
of positive integers, our task is to partition
X
into disjoint subsets
A
and
B
such
that
max
∑
A
,
∑
B
is as small as possible. Solving this problem exactly is clearly
NPhard.
(a) Prove that the following algorithm yields a
(
3
/
2
)
approximation.
G
P
(
X
[
1..
n
])
:
a
←
0
b
←
0
for
i
←
1
to
n
if
a
<
b
a
←
a
+
X
[
i
]
else
b
←
b
+
X
[
i
]
return
max
{
a
,
b
}
(b)
Prove that the approximation ratio
3
/
2
cannot be improved, even if the input
array
X
is sorted in
nondecreasing
order. In other words, give an example of a
sorted array
X
where the output of
G
P
is
50%
larger than the
cost of the optimal partition.
(c)
Prove that if the array
X
is sorted in
nonincreasing
order, then
G
P
achieves an approximation ratio of at most
4
/
3
.
“
(d)
Prove that if the array
X
is sorted in
nonincreasing
order, then
G
P
achieves an approximation ratio of exactly
7
/
6
.
.
The
chromatic number
χ
(
G
)
of a graph
G
is the minimum number of colors required
to color the vertices of the graph, so that every edge has endpoints with different
colors. Computing the chromatic number exactly is NPhard.
Prove that the following problem is also NPhard: Given an
n
vertex graph
G
,
return any integer between
χ
(
G
)
and
χ
(
G
)+
31337
.
[Note: This does not contradict
the possibility of a constant
factor
approximation algorithm.]
J. A
A
.
Let
G
= (
V
,
E
)
be an undirected graph, each of whose vertices is colored either red,
green, or blue. An edge in
G
is
boring
if its endpoints have the same color, and
interesting
if its endpoints have different colors. The
most interesting
coloring
is the