<s>
The	O
term	O
"	O
auction	B-Algorithm
algorithm	I-Algorithm
"	O
applies	O
to	O
several	O
variations	O
of	O
a	O
combinatorial	O
optimization	O
algorithm	O
which	O
solves	O
assignment	B-Algorithm
problems	I-Algorithm
,	O
and	O
network	O
optimization	O
problems	O
with	O
linear	O
and	O
convex/nonlinear	O
cost	O
.	O
</s>
<s>
An	O
auction	B-Algorithm
algorithm	I-Algorithm
has	O
been	O
used	O
in	O
a	O
business	O
setting	O
to	O
determine	O
the	O
best	O
prices	O
on	O
a	O
set	O
of	O
products	O
offered	O
to	O
multiple	O
buyers	O
.	O
</s>
<s>
It	O
is	O
an	O
iterative	O
procedure	O
,	O
so	O
the	O
name	O
"	O
auction	B-Algorithm
algorithm	I-Algorithm
"	O
is	O
related	O
to	O
a	O
sales	O
auction	O
,	O
where	O
multiple	O
bids	O
are	O
compared	O
to	O
determine	O
the	O
best	O
offer	O
,	O
with	O
the	O
final	O
sales	O
going	O
to	O
the	O
highest	O
bidders	O
.	O
</s>
<s>
The	O
original	O
form	O
of	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
iterative	O
method	O
to	O
find	O
the	O
optimal	O
prices	O
and	O
an	O
assignment	O
that	O
maximizes	O
the	O
net	O
benefit	O
in	O
a	O
bipartite	O
graph	O
,	O
the	O
maximum	B-Algorithm
weight	I-Algorithm
matching	I-Algorithm
problem	O
(	O
MWM	O
)	O
.	O
</s>
<s>
The	O
ideas	O
of	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
and	O
ε-scaling	O
are	O
also	O
central	O
in	O
preflow-push	O
algorithms	O
for	O
single	O
commodity	O
linear	O
network	O
flow	O
problems	O
.	O
</s>
<s>
In	O
fact	O
the	O
preflow-push	O
algorithm	O
for	O
max-flow	O
can	O
be	O
derived	O
by	O
applying	O
the	O
original	O
1979	O
auction	B-Algorithm
algorithm	I-Algorithm
to	O
the	O
max	O
flow	O
problem	O
after	O
reformulation	O
as	O
an	O
assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
the	O
preflow-push	O
algorithm	O
for	O
the	O
linear	O
minimum	O
cost	O
flow	O
problem	O
is	O
mathematically	O
equivalent	O
to	O
the	O
ε-relaxation	O
method	O
,	O
which	O
is	O
obtained	O
by	O
applying	O
the	O
original	O
auction	B-Algorithm
algorithm	I-Algorithm
after	O
the	O
problem	O
is	O
reformulated	O
as	O
an	O
equivalent	O
assignment	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
A	O
later	O
variation	O
of	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
that	O
solves	O
shortest	O
path	O
problems	O
was	O
introduced	O
by	O
Bertsekas	O
in	O
1991	O
.	O
</s>
<s>
In	O
the	O
single	O
origin/single	O
destination	O
case	O
,	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
maintains	O
a	O
single	O
path	O
starting	O
at	O
the	O
origin	O
,	O
which	O
is	O
then	O
extended	O
or	O
contracted	O
by	O
a	O
single	O
node	O
at	O
each	O
iteration	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
multiple	O
origins	O
,	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
is	O
well-suited	O
for	O
parallel	O
computation	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
closely	O
related	O
to	O
auction	B-Algorithm
algorithms	I-Algorithm
for	O
other	O
network	O
flow	O
problems	O
.	O
</s>
<s>
According	O
to	O
computational	O
experiments	O
,	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
is	O
generally	O
inferior	O
to	O
other	O
state-of-the-art	O
algorithms	O
for	O
the	O
all	O
destinations	O
shortest	O
path	O
problem	O
,	O
but	O
is	O
very	O
fast	O
for	O
problems	O
with	O
few	O
destinations	O
(	O
substantially	O
more	O
than	O
one	O
and	O
substantially	O
less	O
than	O
the	O
total	O
number	O
of	O
nodes	O
)	O
;	O
see	O
the	O
article	O
by	O
Bertsekas	O
,	O
Pallottino	O
,	O
and	O
Scutella	O
,	O
.	O
</s>
<s>
Auction	B-Algorithm
algorithms	I-Algorithm
for	O
shortest	O
hyperpath	O
problems	O
have	O
been	O
defined	O
by	O
De	O
Leone	O
and	O
Pretolani	O
in	O
1998	O
.	O
</s>
<s>
This	O
is	O
also	O
a	O
parallel	O
auction	B-Algorithm
algorithm	I-Algorithm
for	O
weighted	O
bipartite	O
matching	O
,	O
described	O
by	O
E	O
.	O
Jason	O
Riedy	O
in	O
2004	O
.	O
</s>
<s>
The	O
(	O
sequential	O
)	O
auction	B-Algorithm
algorithms	I-Algorithm
for	O
the	O
shortest	O
path	O
problem	O
have	O
been	O
the	O
subject	O
of	O
experiments	O
which	O
have	O
been	O
reported	O
in	O
technical	O
papers	O
.	O
</s>
<s>
Experiments	O
clearly	O
show	O
that	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
is	O
inferior	O
to	O
the	O
state-of-the-art	O
shortest-path	O
algorithms	O
for	O
finding	O
the	O
optimal	O
solution	O
of	O
single-origin	O
to	O
all-destinations	O
problems	O
.	O
</s>
<s>
Although	O
with	O
the	O
auction	B-Algorithm
algorithm	I-Algorithm
the	O
total	O
benefit	O
is	O
monotonically	O
increasing	O
with	O
each	O
iteration	O
,	O
in	O
the	O
Hungarian	B-Algorithm
algorithm	I-Algorithm
(	O
from	O
Kuhn	O
,	O
1955	O
;	O
Munkres	O
,	O
1957	O
)	O
the	O
total	O
benefit	O
strictly	O
increases	O
with	O
each	O
iteration	O
.	O
</s>
<s>
The	O
auction	B-Algorithm
algorithm	I-Algorithm
of	O
Bertsekas	O
for	O
finding	O
shortest	O
paths	O
within	O
a	O
directed	O
graph	O
is	O
reputed	O
to	O
perform	O
very	O
well	O
on	O
random	O
graphs	O
and	O
on	O
problems	O
with	O
few	O
destinations	O
.	O
</s>
