3
Hardness of Approximation
Theorem 3
The decision version of
BCP
2
is
NP
complete in the strong sense for 2
connected graphs.
Proof:
We will show the reduction from
X
3
C
problem to
BCP
2
problem
The Construction:
Given an instance
(
X,C
)
of
X
3
C
, let
G
= (
V,E
)
be the graph with vertex set
V
=
X
∪
C
∪
a,b
and edge set
E
=
u
3
q
j
=1
[
C
j
x
i

x
i
∈
C
j
∪
C
j
a
∪
C
j
b
]
. Clearly,
G
can be constructed in polynomial time in the size of
(
X,C
)
. It is also not difficult to
see that
G
is 2connected.
w(a)=9q^2+2q
w(b)=3q
w(C_j)=1
w(x_i)=3q
Figure 1
: Example: Graph obtained by the reduction for the instance
(
X,C
)
, where
C
=
{
C
1
,C
2
,...,C
6
}
,
C
1
=
{
x
3
,x
4
,x
5
}
,
C
2
=
{
x
1
,x
2
,x
3
}
,
C
3
=
{
x
1
,x
3
,x
6
}
,
C
4
=
{
x
1
,x
4
,x
6
}
,
C
5
=
{
x
2
,x
5
,x
6
}
and
C
6
=
{
x
2
,x
4
,x
5
}
Define a weight function
w
:
V
→
Z
+
as follows:
w
(
a
) = 9
q
2
+ 2
q
;
w
(
b
) = 3
q
;
w
(
x
i
) = 3
q
for
i
= 1
,...,
3
q
; and
w
(
C
j
) = 1
for
j
= 1
,...,
3
q
. Note that
w
(
V
) =
2(9
q
2
+ 4
q
)
.
We will prove that