<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
operations	O
research	O
,	O
the	O
envy	B-Algorithm
minimization	I-Algorithm
problem	O
is	O
the	O
problem	O
of	O
allocating	O
discrete	O
items	O
among	O
agents	O
with	O
different	O
valuations	O
over	O
the	O
items	O
,	O
such	O
that	O
the	O
amount	O
of	O
envy	O
is	O
as	O
small	O
as	O
possible	O
.	O
</s>
<s>
In	O
general	O
,	O
this	O
optimization	O
problem	O
is	O
NP-hard	O
,	O
since	O
even	O
deciding	O
whether	O
an	O
envy-free	O
allocation	O
exists	O
is	O
equivalent	O
to	O
the	O
partition	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
number	O
of	O
envy	O
relations	O
(	O
-	O
edges	O
in	O
the	O
envy	B-Algorithm
graph	I-Algorithm
)	O
;	O
</s>
<s>
There	O
is	O
a	O
PTAS	B-Algorithm
for	O
max-envy-ratio	O
minimization	O
.	O
</s>
<s>
Furthermore	O
,	O
when	O
the	O
number	O
of	O
players	O
is	O
constant	O
,	O
there	O
is	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
When	O
the	O
number	O
of	O
agents	O
is	O
constant	O
(	O
not	O
a	O
part	O
of	O
the	O
input	O
)	O
,	O
there	O
is	O
an	O
FPTAS	B-Algorithm
for	O
minimizing	O
the	O
maximum	O
envy-ratio	O
,	O
and	O
the	O
product	O
of	O
envy-ratios	O
.	O
</s>
<s>
The	O
problem	O
can	O
be	O
solved	O
using	O
the	O
following	O
local	B-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
After	O
receiving	O
envy	O
evaluations	O
from	O
other	O
agents	O
,	O
the	O
agent	O
may	O
decide	O
to	O
backtrack	B-Algorithm
on	O
a	O
variable	O
;	O
if	O
there	O
are	O
no	O
more	O
variables	O
to	O
backtrack	B-Algorithm
,	O
the	O
agent	O
may	O
backtrack	B-Algorithm
to	O
a	O
previous	O
agent	O
.	O
</s>
<s>
Other	O
settings	O
in	O
which	O
envy	B-Algorithm
minimization	I-Algorithm
was	O
studied	O
are	O
:	O
</s>
