Preemptive Coordination Mechanisms for Unrelated Machines

Preemptive Coordination Mechanisms for Unrelated

Machines

By

Fidaa AbedMax-Planck-Institut für Informatik

Chien-Chung HuangHumboldt-Universität zu Berlin

2

Unrelated Machine Scheduling

Classical problem

m unrelated machines

n jobs

pij – processing time of job i on

machine j

Each job is owned by a selfish

user

User goal: minimize his

completion time

System goal: minimize the

worst completion time

(Makespan)

Job1

Job2

Job3

M1 M2 M3

Job 2 Job 1

Job 3

1

1

1

3 3

3

3

3

3

3

1

3

Scheduling Policies

3

5

25

3

2Longest first

2

3

5Shortest first

4

The Game

Nash equilibrium (NE): no user wants to change his machine

NE can be far from optimal

Cost (NE) = 3Cost (OPT) = 1

PoA = Cost (Worst NE)/Cost (OPT)PoA = 3/1

PoA can be unbounded

Job 1Job 3

Job1

Job2

Job3

M1 M2 M3

1

1

1

3 3

3

3

Job 1

3

3

3

Job 3Job 21

Job 2

Job 1

Longest first

5

Coordination Mechanisms

Clever scheduling policies

Examples:Longest firstShortest first

Can be preemptiveJob1

Job2

Job1

Job2

t1

t2

Job2

Job1

t2

t1

Job2

Job1

Job1t1

t2

6

Goal: Minimize PoA

Shortest firstJob 3Job 2

Job1

Job2

Job3

M1 M2 M3

1

1

1

3 3

3

3

3

3

Job 1

Job 2Job 1 Job 3

Job 1

7

History of Coordination Mechanisms

Introduced by Christodoulou et. al in 2004

Mechanism PoA Preemptive?

Longest first Unbounded No

Shortest first m No

Inefficiency based

logm No

BCOORDlogm

loglogmYes

8

Open Problem

Can we achieve Constant PoA using preemption or randomization?

Azar, Jain, and Mirrokni (SODA 2008)

Caragiannis (SODA 2009)

9

Our Results(1)

• All deterministic mechanisms, even with preemption, if they are

– symmetric– satisfy Independence of Irrelevant Alternatives

(IIA) property

have the PoA Ω ( ) .log mloglog

m

10

Our Results(2)

• All randomized mechanisms, even with preemption, if they are

– symmetric– unbiased

have the PoA Ω ( ) .log mloglog

m

11

Symmetry

t_z

x

yy

t_z

t_y t_y

a b

x

zz

t_xt_xAll known mechanisms are symmetric.

12

Independence of Irrelevant Alternatives (IIA) Property

• If job z is “preferred” over job y by machine a, then this “preference” should not change because of the availability of some other job x.

x

y

t1(z)

t2(y)

z

t3(x)

• appears as axiom in voting theory and logic

• was assumed by Azar et. al. [SODA 2008] in their lower bound

• All known mechanisms have this property

13

IIA Property

Lemma: IIA each machine has order over the jobs

- Proof is omitted.

The order based on:

1. Jobs IDs (non-anonymous case)

2. Machines IDs (anonymous case)

14

Anonymous Case

1 1

1 1

1 1

M4

M2 M3 M4M1

Job

Job

Job

M3 M1 M2

15

Lower Bound for Non-Anonymous Case

node = machine

Edge = job that can go to two machines

Processing time of all jobs = 1

For k =3

k

k-1

m = 1+k + k(k-1) + …… + k!

k = Θ( )

log mloglog

m

16

Lower Bound for Non-Anonymous Case

Cost (OPT) = 1

Cost (NE) >= k

PoA >= k

k = Ω ( )

PoA = Ω ( )log mloglog

m

log mloglog

m

17

Lower Bound for Non-Anonymous Case

x

x

y

z

w

y

z

wxy

yxz

zxyw

afmlkqponjwqgex

18

Lower Bound for Non-Anonymous Case

ab

t_z

x

zy

t_y

t_x t_x x

y

z

a b

19

Lower Bound for Anonymous Case

m1

m2

m3

m1m3 m2

20

Lower Bound for Anonymous Case

21

Lower Bound for Anonymous Case

ab

t_z

x

zy

t_y

t_x t_x x

y

z

a b

22

Positive Result

• The previous lower bound was because of the unbounded inefficiency.

• If the inefficiency is bounded by a constant C then we can achieve constant PoA by known mechanisms.

• Ex: Inefficiency-based mechanism achieves PoA <= C + 2 log C + 2 = O(C).

23

Open Problem

• BCOORD is optimal but it is not known whether it guarantees Pure Nash Equilibrium.

• Open Problem: Design a mechanisms that achieves Θ( ) and guarantees the convergence to Pure Nash Equilibrium.

log mloglog

m

24

Conclusion

• Achieving constant PoA using preemption or randomization is impossible.

• If the inefficiency is bounded then we can achieve constant PoA.