Transcript Document

An Optimal Lower Bound for
Anonymous Scheduling Mechanisms
Ron Lavi
Industrial Engineering and Management
Technion - Israel Institute of Technology
Itai Ashlagi (Harvard Business Scool) and
Shahar Dobzinski (CS, Hebrew University)
Joint Work with
Job Scheduling (example)
two workers(M1, M2);
three tasks (J1, J2, J3):
J1
J2
J3
M1
2
2
3
M2
1
3
4
Need to assign tasks to workers.
A possible assignment:
J2
J3
J1
1
2
Job Scheduling (example)
two workers(M1, M2);
three tasks (J1, J2, J3):
J1
J2
J3
M1
2
2
3
M2
1
3
4
Need to assign tasks to workers.
A possible assignment:
J2
J3
J1
“makespan” = 4
1
2
Job Scheduling (definition)
• n tasks (“jobs”) to be assigned to m workers (“machines”)
• Each machine, i, needs cij time units to complete job j.
• Our goal: to assign jobs to machines to complete all jobs as
soon as possible. More formally:
– Let Si denote the set of jobs assigned to machine i, and
define the load of a machine: li = jSi cij.
– Our goal is then to minimize the maximal load (a.k.a the
“makespan” of the schedule).
Scheduling and Mechanism Design
• Nisan and Ronen (GEB, ‘01): Workers/machines are selfish
entities, each one is acting to maximize her individual utility.
• If job j is assigned to machine i, it will incur a cost cij for
executing the job. cij is private information to machine i.
• A machine may get a payment, Pi, to balance its cost,
and its total utility is: Pi - li
• A truthful mechanism:
– Machines need to report types
– Truthful reporting is a dominant strategy.
Question
• Question: design a truthful mechanism that will reach a “close
to optimal” makespan.
• Approximation ratio: worst ratio (over all instances) of the
mechanism’s makespan to the optimal makespan.
• The “usual” tool: the Vickrey-Clarke-Groves (VCG) method.
Fits cases where we wish to maximize the social welfare.
• Basic observation [Nisan-Ronen]: makespan minimization is
inherently different than welfare maximization, hence VCG
performs poorly (obtains makespan of up to m times the
optimum, i.e. has an approximation ratio of m).
Example
two workers(M1, M2);
three tasks (J1, J2, J3):
J1
J2
J3
M1
2
2
3
M2
1
3
4
Need to assign tasks to workers.
A possible assignment:
J2
J3
“makespan” = 4
J1
Welfare = -3 - 3 -1 = -7
1
2
Example
two workers(M1, M2);
three tasks (J1, J2, J3):
J1
J2
J3
M1
2
2
3
M2
1
3
4
A different assignment:
J2
J3
J1
Makespan = 5
Tot. Welfare = -2 - 3 -1 = -6
1
2
Why is this question important? (1)
• Significant to several disciplines:
– Computer Science
– Operations Research
• Makespan minimization is similar to a Rawls’ max-min
criteria -- gives a justification from social choice theory.
– The implicit goal: assign tasks to workers in a fair
manner (rather than in a socially efficient manner).
– Can we do it via mechanism design?
Why is this question important? (2)
• The general status of mechanism design for multidimensional domains is still unclear.
– What social choice functions can be implemented?
– Few possibilities, few impossibilities, more questions
than answers.
• Scheduling is a multi-dimensional domain, and is
becoming one of the important domains for which we need
to determine the possibilities - impossibilities border.
Current status: special cases
Case I: related machines
[Archer and Tardos (2001)]
machine i has speed si, and cij = cj/si
• Optimal truthful mechanism exists (requires exponential computation).
• Many truthful approximations with polynomial computation:
– A randomized PTAS (Dhangwatnotai, Dobzinski, Dughmi, and Roughgarden ‘08 )
– Deterministic 3-approximation (Kovacs ‘05)
Case II: two-value jobs
[Lavi and Swamy (2007)]
Each processing time is either high or low, in an unrelated way.
• Randomized 3-approx (exponential computation)
• Deterministic 2-approx (polynomial computation, when lows and highs are equal).
• Extension to a “two-range” domain (Yu, 2009)
Current status: lower bounds
• Nisan and Ronen (1999): Every truthful mechanism obtains
approximation ratio > 2.
• Christodoulou, Koutsoupias, and Vidali (2007): an improved
lower bound (about 2.6).
• Mu’alem and Schapira (2007): a 2-(1/m) lower bound for
randomized mechanisms and truthfulness in expectation.
• No non-trivial truthful approximation (i.e. o(m)) is known!
Conjecture (Nisan and Ronen): VCG provides the best possible
approximation ratio.
(Given the many positive results for the special cases and the very
low lower bounds, skepticism is natural)
A bad instance for VCG
J1
…
Jm
M1
t1
…
t1
M2
t2
…
t2
.
.
.
Mm
Optimal
makespan is t1+e
VCG gives
makespan m·t1
tm
…
tm
t1+e > tm > … > t2 > t1
Our Result
Theorem:
Every anonymous and
truthful mechanism with
a bounded approximation
ratio provides the same
assignment as VCG in
this instance.
Corollary:
VCG obtains the best
approximation ratio
among all truthful and
anonymous mechanisms.
J1
…
Jm
M1
t1
…
t1
M2
t2
…
t2
tm
…
tm
.
.
.
Mm
t1+e > tm > … > t2 > t1
Anonymity
Anonymity: if two machines with distinct costs switch types, the
assigned jobs also switch (i.e. machine names do not matter).
Natural requirement:
• Algorithmic perspective: the classic scheduling algorithms are
anonymous.
• Mechanism design perspective: the mechanisms for the special
cases are anonymous.
• Game theory perspective: anonymous games are an important
and natural class.
Weak monotonicity (W-MON)
• DFN (Lavi, Mu’alem, and Nisan ‘03, Bikhchandani et. al. ‘06):
Fix the declarations of the other machines. Suppose machine i
receives a set S of jobs when declaring ci, and a set S’ of jobs
when declaring c’i, . Then ci(S’) - c’i(S’) > ci(S) - c’i(S)
• Every truthful mechanism satisfies W-MON.
• W-MON is necessary for truthfulness if the domain of types is
convex (Saks and Yu, ‘05; Monderer ‘08).
Example 1
J1
…
Jm
M1
t1
…
t1
M1
t1 - e
M2
t2
…
t2
M2
t2
…
t2
tm
…
tm
.
.
.
Mm
J1
…
Jm
… t1 - e
.
.
.
tm
…
tm
Mm
Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = em ?
Example 1
J1
…
Jm
M1
t1
…
t1
M1
t1 - e
M2
t2
…
t2
M2
t2
…
t2
tm
…
tm
.
.
.
Mm
J1
…
Jm
… t1 - e
.
.
.
tm
…
tm
Mm
Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = em ?
Example 2
J1
J2
J3
J1
J2
J3
M1
x
y
z
M1
x-e
M2
?
?
?
M2
?
?
?
M3
?
?
?
M3
?
?
?
y-e z+e
Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = e2 ?
Example 3
J1
J2
J3
M1
1
1
1
M2
1
1
1
J1
J2
J3
M1
e
1+e
1+e
M2
1
1
1
Remarks:
1. this almost finishes the proof of the lower bound of 2.
2. Nisan and Ronen use “one hop” arguments, similar to this.
3. The other lower bounds use increasingly longer hops.
4. We use an inductive argument that enables us to identify
very long hops that give us the optimal lower bound.
Overview of proof
• Proof is by induction on the number of jobs.
• In this overview: only 3 machines and 3 jobs.
• “main lemma”: in the following
instance, M1 receives all jobs,
where x,y in {t1,}.
• Proof is by induction on number
of ’s. In this overview I will
assume correctness for x=y= 
and for x=t1 and y= , and will
prove the claim for x=y=t1.
J1
J2
J3
M1
x
y

M2
t21 t22 t23
M3
t31 t32 t33
t3j> t2j > t1 >> 
Induction steps
J1
J2
J3
J1
J2
J3
M1
t1
t1
a
M1
t1
t1

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
a
M3
t3
t3
a
J1
J2
J3
M1
t1
t1

M2
t2
t2
M3
t3
t3
J1
J2
J3
M1
t1
t1

a
M2
t2
t2
t2
t3
M3
t3
t3
t3
t3> t2 > t1 >> a >> 
Step 1
J1
J2
J3
M1
t1
t1
a
M2
t2
t2
a
M3
t3
t3
a
This induces a mechanism on
three machines and two jobs
and by the induction
assumption the lowest machine
must get both jobs.
Induction steps
J1
J2
J3
J1
J2
J3
M1
t1
t1
a
M1
t1
t1

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
a
M3
t3
t3
a
J1
J2
J3
M1
t1
t1

M2
t2
t2
M3
t3
t3
J1
J2
J3
M1
t1
t1

a
M2
t2
t2
t2
t3
M3
t3
t3
t3
t3> t2 > t1 >> a >> 
Step 2
J1
Towards
Contradiction:
J2
J3
WMON
J1
J2
J3
M1


2
M1
t1
t1

M2
t2
t2
a
M2
T2
t2
a
M3
t3
t3
a
M3
t3
t3
a
makespan = a >> 2 = optimal makespan.
Thus the mechanism does not provide
a finite approx ratio, a contradiction.
Induction steps
J1
J2
J3
J1
J2
J3
M1
t1
t1
a
M1
t1
t1

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
a
M3
t3
t3
a
J1
J2
J3
M1
t1
t1

M2
t2
t2
M3
t3
t3
J1
J2
J3
M1
t1
t1

a
M2
t2
t2
t2
t3
M3
t3
t3
t3
t3> t2 > t1 >> a >> 
Step 3
Claim 3(a): If M1 receives either J1 or
J2 then it must receive J1 and J2.
Proof:
J1
J2
J3
J1
J2
J3
M1
t1
t1

M1

t1+e

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
Towards a contradiction
A contradiction to
the induction hypothesis of
the “main lemma”
(M1 should get everything)
Step 3
Claim 3(b): If M1 receives J1 and J2 it
must also receive J3.
Proof: (exactly like step 2)
J1
Towards
Contradiction:
J2
J3
WMON
J1
J2
J3
M1


2
M1
t1
t1

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
makespan > a >> 2 = optimal makespan.
Thus the mechanism does not provide
a finite approx ratio, a contradiction.
Step 3
Claim 3(c): M1 receives either J1 or J2.
Proof: otherwise find t1 < t’1 < t’’1 < t2 and  < ‘ << ‘‘ < a such that:
J1
J2 J3
J1
J2
J3
‘‘
M1
t’1 t’1 ‘
M1 t’’1 t’’1
M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
Step 3
Claim 3(c): M1 receives either J1 or J2.
Proof: otherwise find t1 < t’1 < t’’1 < t2 and  < ‘ << ‘‘ < a such that:
J1
J2 J3
J1
J2
J3
‘‘
M1
t’1 t’1 ‘
M1 t’’1 t’’1
M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
By WMON, since t2 - t1 > a.
J1
J2
J3
M1 t’’1 t’’1 ‘‘
M2 t’1
t’1
‘
M3
t3
t3
t3
Step 3
Claim 3(c): M1 receives either J1 or J2.
Proof: otherwise find t1 < t’1 < t’’1 < t2 and  < ‘ << ‘‘ < a such that:
J1
J2 J3
J1
J2
J3
‘‘
M1
t’1 t’1 ‘
M1 t’’1 t’’1
M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
By WMON, since t2 - t1 > a.
J1
J2
J3
M1 t’’1 t’’1 ‘‘
M2 t’1
t’1
‘
M3
t3
t3
t3
J1
J2
J3
M1 t’’1 t’’1 ‘‘
claim M t’
2
1
3(a+b)
M3
t3
t’1
‘
t3
t3
Step 3
Claim 3(c): M1 receives either J1 or J2.
Proof: otherwise find t1 < t’1 < t’’1 < t2 and  < ‘ << ‘‘ < a such that:
J1
J2 J3
J1
J2
J3
‘‘
M1
t’1 t’1 ‘
M1 t’’1 t’’1
M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
By WMON, since t2 - t1 > a.
J1
J1
J2
J3
t2
t2
a
M2 t’1
t’1
M3
t3
‘ WMON M2 t’1
M 3 t3
t3
M1
t3
J2
J3
M1 t’’1 t’’1 ‘‘
t’1
‘
t3
t3
J1
J2
J3
M1 t’’1 t’’1 ‘‘
claim M t’
2
1
3(a+b)
M3
t3
t’1
‘
t3
t3
Step 3
Claim 3(c): M1 receives either J1 or J2.
Proof: otherwise find t1 < t’1 < t’’1 < t2 and  < ‘ << ‘‘ < a such that:
J1
Contradiction
to anonymity
J2 J3
J1
‘‘
t’1 t’1 ‘
M1 t’’1 t’’1
M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
t3
M3
t3
t3
t3
By WMON, since t2 - t1 > a.
J1
J2
J3
t2
t2
a
M2 t’1
t’1
M3
t3
‘ WMON M2 t’1
M 3 t3
t3
t3
J3
M1
J1
M1
J2
J2
J3
M1 t’’1 t’’1 ‘‘
t’1
‘
t3
t3
J1
J2
J3
M1 t’’1 t’’1 ‘‘
claim M t’
2
1
3(a+b)
M3
t3
t’1
‘
t3
t3
Induction steps
J1
J2
J3
J1
J2
J3
M1
t1
t1
a
M1
t1
t1

M2
t2
t2
a
M2
t2
t2
a
M3
t3
t3
a
M3
t3
t3
a
J1
J2
J3
M1
t1
t1

M2
t2
t2
M3
t3
t3
J1
J2
J3
M1
t1
t1

a
M2
t2
t2
t2
t3
M3
t3
t3
t3
t3> t2 > t1 >> a >> 
Final step
“main lemma”
J1
J2
J3
M1
t1
t1
t1
M2
t2
t2
t2
M3
t3
t3
t3
Proof similar to that of step 3:
Claim 1: If M1 receives at least one job then it receives all jobs.
Claim 2: M1 must receive at least one job (otherwise we
construct a contradiction to anonymity)
Summary
• Study scheduling mechanisms to minimize the makespan.
• Result: VCG is the best anonymous and truthful mechanism.
– Negative result, since VCG may output a large makespan.
• Technical method: repeatedly applying WMON to create very
long contradiction paths.
– Instead of proving a characterization result.
• Further directions:
– Can non-anonymous mechanisms do better?
– Can randomized mechanisms do better?
– Perhaps over a discrete domain?
– Perhaps using alternative solution concepts?