<s>
Truthful	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
is	O
a	O
mechanism	O
design	O
variant	O
of	O
the	O
job	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
problem	O
from	O
operations	O
research	O
.	O
</s>
<s>
We	O
have	O
a	O
project	O
composed	O
of	O
several	O
"	O
jobs	B-General_Concept
"	O
(	O
tasks	O
)	O
.	O
</s>
<s>
Our	O
goal	O
is	O
to	O
allocate	O
jobs	B-General_Concept
to	O
workers	O
such	O
that	O
the	O
total	O
makespan	B-Algorithm
of	O
the	O
project	O
is	O
minimized	O
.	O
</s>
<s>
In	O
the	O
standard	O
job	B-Algorithm
shop	I-Algorithm
scheduling	I-Algorithm
problem	O
,	O
the	O
timings	O
of	O
all	O
workers	O
are	O
known	O
,	O
so	O
we	O
have	O
a	O
standard	O
optimization	O
problem	O
.	O
</s>
<s>
In	O
contrast	O
,	O
in	O
the	O
truthful	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
problem	O
,	O
the	O
timings	O
of	O
the	O
workers	O
are	O
not	O
known	O
.	O
</s>
<s>
The	O
truthful	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
problem	O
was	O
introduced	O
by	O
Nisan	O
and	O
Ronen	O
in	O
their	O
1999	O
paper	O
on	O
algorithmic	O
mechanism	O
design	O
.	O
</s>
<s>
There	O
are	O
jobs	B-General_Concept
and	O
workers	O
(	O
"	O
m	O
"	O
stands	O
for	O
"	O
machine	O
"	O
,	O
since	O
the	O
problem	O
comes	O
from	O
scheduling	O
jobs	B-General_Concept
to	O
computers	O
)	O
.	O
</s>
<s>
If	O
worker	O
is	O
assigned	O
a	O
set	O
of	O
jobs	B-General_Concept
,	O
then	O
he	O
can	O
execute	O
them	O
in	O
time	O
:	O
</s>
<s>
Given	O
an	O
allocation	O
of	O
jobs	B-General_Concept
to	O
workers	O
,	O
The	O
makespan	B-Algorithm
of	O
a	O
project	O
is	O
:	O
</s>
<s>
An	O
optimal	O
allocation	O
is	O
an	O
allocation	O
of	O
jobs	B-General_Concept
to	O
workers	O
in	O
which	O
the	O
makespan	B-Algorithm
is	O
minimized	O
.	O
</s>
<s>
The	O
minimum	O
makespan	B-Algorithm
is	O
denoted	O
by	O
.	O
</s>
<s>
An	O
allocation	O
of	O
jobs	B-General_Concept
to	O
workers	O
,	O
;	O
</s>
<s>
The	O
research	O
on	O
truthful	B-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
aims	O
to	O
find	O
upper	O
(	O
positive	O
)	O
and	O
lower	O
(	O
negative	O
)	O
bounds	O
on	O
approximation	O
factors	O
of	O
truthful	O
mechanisms	O
.	O
</s>
<s>
Let	O
OPT	O
be	O
an	O
allocation	O
which	O
minimizes	O
the	O
makespan	B-Algorithm
.	O
</s>
<s>
Suppose	O
there	O
are	O
jobs	B-General_Concept
and	O
the	O
timings	O
of	O
the	O
workers	O
are	O
as	O
follows	O
:	O
</s>
<s>
Both	O
the	O
"	O
make-total	O
"	O
and	O
the	O
makespan	B-Algorithm
are	O
.	O
</s>
<s>
But	O
,	O
when	O
each	O
job	O
is	O
assigned	O
to	O
a	O
different	O
worker	O
,	O
the	O
makespan	B-Algorithm
is	O
.	O
</s>
<s>
In	O
the	O
following	O
proof	O
sketch	O
,	O
for	O
simplicity	O
we	O
assume	O
that	O
there	O
are	O
2	O
workers	O
and	O
that	O
the	O
number	O
of	O
jobs	B-General_Concept
is	O
even	O
,	O
.	O
</s>
<s>
The	O
timings	O
of	O
both	O
workers	O
to	O
all	O
jobs	B-General_Concept
are	O
1	O
.	O
</s>
<s>
Suppose	O
,	O
without	O
loss	O
of	O
generality	O
,	O
that	O
(	O
worker	O
1	O
is	O
assigned	O
at	O
most	O
as	O
many	O
jobs	B-General_Concept
as	O
worker	O
2	O
)	O
.	O
</s>
<s>
The	O
timings	O
of	O
worker	O
1	O
to	O
the	O
jobs	B-General_Concept
in	O
are	O
(	O
a	O
very	O
small	O
positive	O
constant	O
)	O
;	O
the	O
timings	O
of	O
worker	O
1	O
to	O
the	O
jobs	B-General_Concept
in	O
are	O
;	O
and	O
the	O
timings	O
of	O
worker	O
2	O
to	O
all	O
jobs	B-General_Concept
is	O
still	O
1	O
.	O
</s>
<s>
Worker	O
1	O
knows	O
that	O
,	O
if	O
he	O
lies	O
and	O
says	O
that	O
his	O
timings	O
to	O
all	O
jobs	B-General_Concept
are	O
1	O
,	O
the	O
(	O
deterministic	O
)	O
mechanism	O
will	O
allocate	O
him	O
the	O
jobs	B-General_Concept
in	O
and	O
his	O
cost	O
will	O
be	O
very	O
near	O
0	O
.	O
</s>
<s>
However	O
,	O
the	O
makespan	B-Algorithm
can	O
be	O
made	O
half	O
as	O
large	O
by	O
dividing	O
the	O
jobs	B-General_Concept
in	O
equally	O
between	O
the	O
agents	O
.	O
</s>
<s>
Consider	O
the	O
special	O
case	O
of	O
Uniform-machines	B-Algorithm
scheduling	I-Algorithm
,	O
in	O
which	O
the	O
workers	O
are	O
single-parametric	O
:	O
for	O
each	O
worker	O
there	O
is	O
a	O
speed	O
,	O
and	O
the	O
time	O
it	O
takes	O
the	O
worker	O
to	O
do	O
a	O
job	O
is	O
the	O
job	O
length	O
divided	O
by	O
the	O
speed	O
.	O
</s>
<s>
Ambrosio	O
and	O
Auletta	O
proved	O
that	O
the	O
Longest	B-Algorithm
Processing	I-Algorithm
Time	I-Algorithm
algorithm	O
is	O
monotone	O
whenever	O
the	O
machine	O
speeds	O
are	O
powers	O
of	O
some	O
c	O
≥	O
2	O
,	O
but	O
not	O
when	O
c	O
≤	O
1.78	O
.	O
</s>
<s>
In	O
contrast	O
,	O
List	B-Algorithm
scheduling	I-Algorithm
is	O
not	O
monotone	O
for	O
c	O
>	O
2	O
.	O
</s>
