On Designing Truthful Mechanisms for Online Scheduling V. Auletta, R. De Prisco, P.P. and G....
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Transcript of On Designing Truthful Mechanisms for Online Scheduling V. Auletta, R. De Prisco, P.P. and G....
![Page 1: On Designing Truthful Mechanisms for Online Scheduling V. Auletta, R. De Prisco, P.P. and G. Persiano Università di Salerno.](https://reader034.fdocuments.in/reader034/viewer/2022051515/5513d72855034674748b4fad/html5/thumbnails/1.jpg)
On Designing Truthful Mechanisms for
Online Scheduling
V. Auletta, R. De Prisco, P.P. and G. Persiano
Università di Salerno
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The Internet
Open, self organized, no central authority, anarchic
Different “components” which• have their own goal• may not follow the “protocol”
Selfish agents
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The Internet
Open, self organized, no central authority, anarchic
An Autonomous System may report false link status to redirect traffic to another AS
AS1
AS2source destination
Link down
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Routing/Scheduling
•Unsplittable traffic J1, J2 ,…,Jn
•We look at the network congestion (makespan)
source destination
Scheduling Selfish Machines:Selfish users own the links and privately know their speeds
s1
sm
s2
0
0
0
Wi= Jk assigned to machine imax i Wi /si
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Mechanism design
Mechanism: M=(A,P)
Computes a solution
X=A(r1,r2,…, ri ,…,rm )
Provides a payment
Pi(r1,r2,…, ri ,…,rm )
Agents’ GOAL: maximize their own utility ui (ri) := Pi(r1,r2,…, ri ,…,rm ) – costi(X,si)
costi(X,si) = wi/si
s1,s2,…, si ,…,sn
true input
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Mechanism design
Strategyproof mechanisms:
no incentive to lie (report ri si)
ui (si) ui (ri)
(truth-telling is the best strategy)
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Scheduling Selfish Machines
Monotone algorithms: an agent declaring a higher speed does not get less work.
A monotone M=(A,P) strategyproof
[Archer & Tardos, FOCS 2001]
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Example: Greedy Algorithm
1 1+
2
2
(1+)2
1+ 1
32
23
NOT MONOTONE
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Related Work• Algorithms:
• (1 + )-APX for any m [Hochbaum & Shmoys, J. ACM 1987]
• 8-competitive for any m [Aspnes & Azar & Fiat & Plotkin & Waarts, STOC93]
• -competitive for m = 2 [Graham, Bell Syst. J. 1966], no better than 3/2
• Monotone Algorithms (Mechanisms):• 5-APX for any m [Andelman & Azar & Sorani, STACS05]
• (1 + )-APX for m = O(1) [Andelman & Azar & Sorani, STACS05]
• Mechanisms With Verification:• (1 + )-APX for any m [Auletta & De Prisco & P. & Persiano, ICALP04]
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Monotonization techniques
AmonA M=(Amon,P)
Algorithm Mechanism
M=(A,P)A
hard
loss of performance
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Our contribution (1/2)
A black-box, polytime
AmonA“easy”
c-apx c’-apxc’ = c(1+)
c • < c’ c •
Offline:
Online: Jobs arrive one by one, no reallocation!
must loose something
AmonAhard
c-comp c’-comp
(the case of two machines)
Proved for any m = O(1) in [Andelman & Azar & Sorani, STACS05]
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Lower Bound
Theorem 3: There is no r-competitive online monotone algorithm, where r min {r, 1+1/r} and r > 1
Corollary 4: No truthful mechanism can be less than -competitive, even for 2 jobs and 2 machines.
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Lower Bound
r
1 r
Theorem 3: There is no r-competitive online monotone algorithm, where r min {r, 1+1/r} and r > 1
11 1
Proof:1 r
1r
r/opt = r/1=r
rr
1+1/r
1
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Upper Bound (m = 2)Theorem 5: AmonA
c-comp c’-comp
• c’ max {cr, 1+1/r}, r 1 • c is the comp. ratio on identical speeds
Corollary 5: GreedymonGreedy3/2-comp c’-comp
c’= 1.823...
Lower bound: =1.62… (2 jobs).
(1-comp) (-comp)
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Our contribution (2/2)(any number of machines)
Mechanisms with Verification: Observe jobs’ released time
Weak Monotonicity Suffices [Auletta et al, ICALP04]
Online 12-competitive strategyproof mechanism for any m
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w1 w2 … wi … wm
sis1 s2 sm
w1’ w2’ …wi’
… wm’
si’s1 s2 sm
wi’ > 0
si’ > si
wi > 0
Weakly Monotone algorithms:
Mechanisms With Verification
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Mechanisms With Verification
w1 w2
wi wm
sis1 s2 sm
An 8-competitive algorithm:
2opt
… …
Jk
JkJk
JkJkJk
Try UB = 1, 2, 4, 8, ... stop UB ≤ 2opt
UB
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Mechanisms With Verification
s1 smsi
Problem: Jk
“hole”sj>si’>si+1
sj+1machine shifts
sj
Jk
no work
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Mechanisms With Verification
Fix: Avoid “Holes”
JkJk
Jk Jk
OK NO, Reallocate
JkJk
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Mechanisms With Verification
Fix: Analysis
Jk
Original alg
Reallocated
8opt
4opt
12-comp. mechanism
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Open Questions
• Close the gaps:
2 1.62… 1.823…
O(1) 1.62… ????
m Lower Bound Upper Bound
No verification
any 1.62… 12Verification
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Thank You