<s>
The	O
Christofides	B-Algorithm
algorithm	I-Algorithm
or	O
Christofides	O
–	O
Serdyukov	O
algorithm	O
is	O
an	O
algorithm	O
for	O
finding	O
approximate	O
solutions	O
to	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
,	O
on	O
instances	O
where	O
the	O
distances	O
form	O
a	O
metric	O
space	O
(	O
they	O
are	O
symmetric	O
and	O
obey	O
the	O
triangle	O
inequality	O
)	O
.	O
</s>
<s>
It	O
is	O
an	O
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
guarantees	O
that	O
its	O
solutions	O
will	O
be	O
within	O
a	O
factor	O
of	O
3/2	O
of	O
the	O
optimal	O
solution	O
length	O
,	O
and	O
is	O
named	O
after	O
Nicos	O
Christofides	O
and	O
Anatoliy	O
I	O
.	O
Serdyukov	O
,	O
who	O
discovered	O
it	O
independently	O
in	O
1976	O
.	O
</s>
<s>
This	O
algorithm	O
still	O
stands	O
as	O
the	O
best	O
polynomial	O
time	O
approximation	B-Algorithm
algorithm	I-Algorithm
that	O
has	O
been	O
thoroughly	O
peer-reviewed	O
by	O
the	O
relevant	O
scientific	O
community	O
for	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
on	O
general	O
metric	O
spaces	O
.	O
</s>
<s>
In	O
July	O
2020	O
however	O
,	O
Karlin	O
,	O
Klein	O
,	O
and	O
Gharan	O
released	O
a	O
preprint	O
in	O
which	O
they	O
introduced	O
a	O
novel	O
approximation	B-Algorithm
algorithm	I-Algorithm
and	O
claimed	O
that	O
its	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
1.510	O
−36	O
.	O
</s>
<s>
Let	O
be	O
an	O
instance	O
of	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
algorithm	O
can	O
be	O
described	O
in	O
pseudocode	B-Language
as	O
follows	O
.	O
</s>
<s>
Combine	O
the	O
edges	O
of	O
and	O
to	O
form	O
a	O
connected	O
multigraph	B-Language
in	O
which	O
each	O
vertex	O
has	O
even	O
degree	O
.	O
</s>
<s>
There	O
exist	O
inputs	O
to	O
the	O
travelling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
that	O
cause	O
the	O
Christofides	B-Algorithm
algorithm	I-Algorithm
to	O
find	O
a	O
solution	O
whose	O
approximation	B-Algorithm
ratio	I-Algorithm
is	O
arbitrarily	O
close	O
to	O
3/2	O
.	O
</s>
<s>
Hence	O
we	O
obtain	O
an	O
approximation	B-Algorithm
ratio	I-Algorithm
of	O
3/2	O
.	O
</s>
