<s>
In	O
network	O
theory	O
,	O
the	O
Wiener	B-Algorithm
connector	I-Algorithm
is	O
a	O
means	O
of	O
maximizing	O
efficiency	O
in	O
connecting	O
specified	O
"	O
query	O
vertices	O
"	O
in	O
a	O
network	O
.	O
</s>
<s>
Given	O
a	O
connected	O
,	O
undirected	O
graph	O
and	O
a	O
set	O
of	O
query	O
vertices	O
in	O
a	O
graph	O
,	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
is	O
an	O
induced	O
subgraph	O
that	O
connects	O
the	O
query	O
vertices	O
and	O
minimizes	O
the	O
sum	O
of	O
shortest	O
path	O
distances	O
among	O
all	O
pairs	O
of	O
vertices	O
in	O
the	O
subgraph	O
.	O
</s>
<s>
In	O
combinatorial	O
optimization	O
,	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
is	O
the	O
problem	O
of	O
finding	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
was	O
first	O
presented	O
by	O
Ruchansky	O
et	O
al	O
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
has	O
applications	O
in	O
many	O
domains	O
where	O
there	O
is	O
a	O
graph	O
structure	O
and	O
an	O
interest	O
in	O
learning	O
about	O
connections	O
between	O
sets	O
of	O
individuals	O
.	O
</s>
<s>
The	O
Wiener	B-Algorithm
connector	I-Algorithm
was	O
named	O
in	O
honor	O
of	O
chemist	O
Harry	O
Wiener	O
who	O
first	O
introduced	O
the	O
Wiener	O
Index	O
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
is	O
related	O
to	O
the	O
Steiner	O
tree	O
problem	O
.	O
</s>
<s>
In	O
fact	O
,	O
a	O
solution	O
for	O
the	O
Steiner	O
tree	O
problem	O
may	O
be	O
arbitrarily	O
bad	O
for	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
;	O
the	O
graph	O
on	O
the	O
right	O
provides	O
an	O
example	O
.	O
</s>
<s>
The	O
problem	O
is	O
NP-hard	O
,	O
and	O
does	O
not	O
admit	O
a	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
Although	O
there	O
is	O
no	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
,	O
there	O
is	O
a	O
polynomial-time	O
constant-factor	O
approximation	O
—	O
an	O
algorithm	O
that	O
finds	O
a	O
connector	O
whose	O
Wiener	O
index	O
is	O
within	O
a	O
constant	O
multiplicative	O
factor	O
of	O
the	O
Wiener	O
index	O
of	O
the	O
optimum	O
connector	O
.	O
</s>
<s>
In	O
terms	O
of	O
complexity	O
classes	O
,	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
is	O
in	O
APX	B-Algorithm
but	O
is	O
not	O
in	O
PTAS	O
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
There	O
is	O
a	O
constant-factor	B-Algorithm
approximation	I-Algorithm
algorithm	I-Algorithm
for	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
that	O
runs	O
in	O
time	O
on	O
a	O
graph	O
with	O
n	O
vertices	O
,	O
m	O
edges	O
,	O
and	O
q	O
query	O
vertices	O
,	O
roughly	O
the	O
same	O
time	O
it	O
takes	O
to	O
compute	O
shortest-path	O
distances	O
from	O
the	O
query	O
vertices	O
to	O
every	O
other	O
vertex	O
in	O
the	O
graph	O
.	O
</s>
<s>
The	O
central	O
approach	O
of	O
this	O
algorithm	O
is	O
to	O
reduce	O
the	O
problem	O
to	O
the	O
vertex-weighted	O
Steiner	O
tree	O
problem	O
,	O
which	O
admits	O
a	O
constant-factor	O
approximation	O
in	O
particular	O
instances	O
related	O
to	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
problem	O
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
behaves	O
like	O
betweenness	O
centrality	O
.	O
</s>
<s>
When	O
the	O
query	O
vertices	O
belong	O
to	O
the	O
same	O
community	O
,	O
the	O
non-query	O
vertices	O
that	O
form	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
tend	O
to	O
belong	O
to	O
the	O
same	O
community	O
and	O
have	O
high	O
centrality	O
within	O
the	O
community	O
.	O
</s>
<s>
When	O
the	O
query	O
vertices	O
belong	O
to	O
different	O
communities	O
,	O
the	O
non-query	O
vertices	O
that	O
form	O
the	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
contain	O
vertices	O
adjacent	O
to	O
edges	O
that	O
bridge	O
the	O
different	O
communities	O
.	O
</s>
<s>
The	O
minimum	O
Wiener	B-Algorithm
connector	I-Algorithm
is	O
useful	O
in	O
applications	O
in	O
which	O
one	O
wishes	O
to	O
learn	O
about	O
the	O
relationship	O
between	O
a	O
set	O
of	O
vertices	O
in	O
a	O
graph	O
.	O
</s>
<s>
in	O
social	O
networks	O
(	O
like	O
Twitter	B-Application
)	O
,	O
it	O
demonstrates	O
the	O
communities	O
to	O
which	O
a	O
set	O
of	O
users	O
belong	O
and	O
how	O
these	O
communities	O
are	O
related	O
,	O
</s>
<s>
in	O
computer	B-Architecture
networks	I-Architecture
,	O
it	O
may	O
be	O
useful	O
in	O
identifying	O
an	O
efficient	O
way	O
to	O
route	O
a	O
multicast	O
message	O
to	O
a	O
set	O
of	O
destinations	O
.	O
</s>
