On Designing Truthful Mechanisms for
Online Scheduling
V. Auletta, R. De Prisco, P.P. and G. Persiano
Università di Salerno
The Internet
Open, self organized, no central authority, anarchic
Different “components” which• have their own goal• may not follow the “protocol”
Selfish agents
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
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
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
Mechanism design
Strategyproof mechanisms:
no incentive to lie (report ri si)
ui (si) ui (ri)
(truth-telling is the best strategy)
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]
Example: Greedy Algorithm
1 1+
2
2
(1+)2
1+ 1
32
23
NOT MONOTONE
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]
Monotonization techniques
AmonA M=(Amon,P)
Algorithm Mechanism
M=(A,P)A
hard
loss of performance
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]
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.
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
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)
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
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
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
Mechanisms With Verification
s1 smsi
Problem: Jk
“hole”sj>si’>si+1
sj+1machine shifts
sj
Jk
no work
Mechanisms With Verification
Fix: Avoid “Holes”
JkJk
Jk Jk
OK NO, Reallocate
JkJk
Mechanisms With Verification
Fix: Analysis
Jk
Original alg
Reallocated
8opt
4opt
12-comp. mechanism
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
Thank You