<s>
Rental	B-Algorithm
harmony	I-Algorithm
is	O
a	O
kind	O
of	O
a	O
fair	O
division	O
problem	O
in	O
which	O
indivisible	O
items	O
and	O
a	O
fixed	O
monetary	O
cost	O
have	O
to	O
be	O
divided	O
simultaneously	O
.	O
</s>
<s>
The	O
housemates	B-Algorithm
problem	I-Algorithm
and	O
room-assignment-rent-division	O
are	O
alternative	O
names	O
to	O
the	O
same	O
problem	O
.	O
</s>
<s>
Su	O
's	O
protocol	O
operates	O
on	O
a	O
dualized	O
version	O
of	O
this	O
simplex	O
in	O
a	O
similar	O
way	O
to	O
the	O
Simmons	B-Algorithm
–	I-Algorithm
Su	I-Algorithm
protocols	I-Algorithm
for	O
cake-cutting	O
:	O
for	O
every	O
vertex	O
of	O
a	O
triangulation	O
of	O
the	O
dual	O
simplex	O
,	O
which	O
corresponds	O
to	O
a	O
certain	O
price	O
scheme	O
,	O
it	O
asks	O
the	O
owning	O
partner	O
"	O
which	O
room	O
do	O
you	O
prefer	O
in	O
that	O
pricing	O
scheme	O
?	O
"	O
.	O
</s>
<s>
Su	O
's	O
Rental	B-Algorithm
Harmony	I-Algorithm
protocol	O
has	O
been	O
popularized	O
in	O
several	O
news	O
articles	O
,	O
and	O
has	O
several	O
online	O
implementations	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
a	O
maxsum	O
allocation	O
is	O
known	O
as	O
the	O
assignment	B-Algorithm
problem	I-Algorithm
,	O
and	O
it	O
can	O
be	O
solved	O
by	O
the	O
Hungarian	B-Algorithm
algorithm	I-Algorithm
in	O
time	O
(	O
where	O
is	O
the	O
number	O
of	O
partners	O
)	O
.	O
</s>
<s>
Such	O
price-vector	O
is	O
a	O
solution	O
of	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
,	O
and	O
it	O
can	O
be	O
found	O
by	O
the	O
Bellman	B-Algorithm
–	I-Algorithm
Ford	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Therefore	O
they	O
build	O
an	O
algorithmic	O
framework	O
,	O
based	O
on	O
linear	B-Algorithm
programming	I-Algorithm
,	O
for	O
calculating	O
allocations	O
that	O
are	O
both	O
envy-free	O
and	O
optimize	O
some	O
criterion	O
.	O
</s>
