<s>
The	O
envy-graph	B-Algorithm
procedure	I-Algorithm
(	O
also	O
called	O
the	O
envy-cycles	B-Algorithm
procedure	I-Algorithm
)	O
is	O
a	O
procedure	O
for	O
fair	O
item	O
allocation	O
.	O
</s>
<s>
The	O
envy-graph	B-Algorithm
procedure	I-Algorithm
aims	O
to	O
achieve	O
the	O
"	O
next-best	O
"	O
option	O
--	O
envy-freeness	O
up	O
to	O
at	O
most	O
a	O
single	O
good	O
(	O
EF1	O
)	O
:	O
it	O
finds	O
an	O
allocation	O
in	O
which	O
the	O
envy	O
of	O
every	O
person	O
towards	O
every	O
other	O
person	O
is	O
bounded	O
by	O
the	O
maximum	O
marginal	O
utility	O
it	O
derives	O
from	O
a	O
single	O
item	O
.	O
</s>
<s>
The	O
envy-graph	B-Algorithm
procedure	I-Algorithm
assumes	O
that	O
each	O
person	O
has	O
a	O
cardinal	O
utility	O
function	O
on	O
bundles	O
of	O
items	O
.	O
</s>
<s>
depth-first	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
The	O
envy-graph	B-Algorithm
algorithm	I-Algorithm
guarantees	O
EF1	O
when	O
the	O
items	O
are	O
goods	O
(	O
-	O
the	O
marginal	O
value	O
of	O
each	O
item	O
is	O
positive	O
for	O
all	O
agents	O
)	O
.	O
</s>
<s>
When	O
agents	O
have	O
cardinality	O
constraints	O
(	O
i.e.	O
,	O
for	O
each	O
category	O
of	O
items	O
,	O
there	O
is	O
an	O
upper	O
bound	O
on	O
the	O
number	O
of	O
items	O
each	O
agent	O
an	O
get	O
from	O
this	O
category	O
)	O
,	O
the	O
envy-graph	B-Algorithm
algorithm	I-Algorithm
might	O
fail	O
.	O
</s>
<s>
However	O
,	O
combining	O
it	O
with	O
the	O
round-robin	B-Algorithm
protocol	I-Algorithm
gives	O
an	O
algorithm	O
that	O
finds	O
allocations	O
that	O
are	O
both	O
EF1	O
and	O
satisfy	O
the	O
cardinality	O
constraints	O
.	O
</s>
<s>
When	O
the	O
agents	O
have	O
assignment	O
valuations	O
(	O
aka	O
OXS	O
valuations	O
)	O
,	O
there	O
is	O
an	O
extension	O
of	O
the	O
envy-graph	B-Algorithm
algorithm	I-Algorithm
called	O
"	O
Algorithm	O
H	O
"	O
,	O
in	O
which	O
the	O
next	O
allocation	O
to	O
an	O
unenvied	O
agent	O
is	O
selected	O
such	O
that	O
agent-item	O
utility	O
is	O
maximized	O
.	O
</s>
