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