<s>
The	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
,	O
also	O
called	O
Bellman	O
–	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
,	O
is	O
a	O
dynamic	B-Algorithm
programming	I-Algorithm
algorithm	O
proposed	O
in	O
1962	O
independently	O
by	O
Bellman	O
and	O
by	O
Held	O
and	O
Karp	O
to	O
solve	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
(	O
TSP	O
)	O
,	O
in	O
which	O
the	O
input	O
is	O
a	O
distance	O
matrix	O
between	O
a	O
set	O
of	O
cities	O
,	O
and	O
the	O
goal	O
is	O
to	O
find	O
a	O
minimum-length	O
tour	O
that	O
visits	O
each	O
city	O
exactly	O
once	O
before	O
returning	O
to	O
the	O
starting	O
point	O
.	O
</s>
<s>
The	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
begins	O
by	O
calculating	O
,	O
for	O
each	O
set	O
of	O
cities	O
and	O
every	O
city	O
not	O
contained	O
in	O
,	O
the	O
shortest	O
one-way	O
path	O
from	O
to	O
that	O
passes	O
through	O
every	O
city	O
in	O
in	O
some	O
order	O
(	O
but	O
not	O
through	O
any	O
other	O
cities	O
)	O
.	O
</s>
<s>
The	O
Held	B-Algorithm
–	I-Algorithm
Karp	I-Algorithm
algorithm	I-Algorithm
has	O
exponential	O
time	O
complexity	O
,	O
significantly	O
better	O
than	O
the	O
superexponential	O
performance	O
of	O
a	O
brute-force	O
algorithm	O
.	O
</s>
<s>
This	O
is	O
analogous	O
to	O
a	O
bidirectional	B-Algorithm
search	I-Algorithm
starting	O
at	O
and	O
meeting	O
at	O
midpoint	O
.	O
</s>
<s>
Besides	O
Dynamic	B-Algorithm
Programming	I-Algorithm
,	O
Linear	B-Algorithm
programming	I-Algorithm
and	O
Branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
are	O
design	O
patterns	O
also	O
used	O
for	O
exact	O
solutions	O
to	O
the	O
TSP	O
.	O
</s>
<s>
Linear	B-Algorithm
programming	I-Algorithm
applies	O
the	O
cutting	O
plane	O
method	O
in	O
integer	B-Algorithm
programming	I-Algorithm
,	O
i.e.	O
</s>
<s>
The	O
term	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
was	O
first	O
used	O
in	O
1963	O
in	O
a	O
paper	O
published	O
by	O
Little	O
et	O
al	O
.	O
</s>
<s>
These	O
algorithms	O
include	O
:	O
Interpolation	O
algorithm	O
,	O
Nearest	B-Algorithm
neighbour	I-Algorithm
algorithm	I-Algorithm
,	O
Clark	O
&	O
Wright	O
algorithm	O
,	O
Double	O
spanning	O
tree	O
algorithm	O
,	O
Christofides	B-Algorithm
algorithm	I-Algorithm
,	O
Hybrid	O
algorithm	O
,	O
Probabilistic	O
algorithm	O
(	O
such	O
as	O
Simulated	B-Algorithm
annealing	I-Algorithm
)	O
.	O
</s>
