Preemptive Coordination Mechanisms for Unrelated Machines

1. Preemptive Coordination Mechanisms for Unrelated Machines By Fidaa Abed Max-Planck-Institut für Informatik Chien-Chung Huang Humboldt-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) 3 Job 2 Job 1 1 Job 3 M1 M2 M3 1 3 3 Job1 3 1 3 Job2 3 3 1 Job3

3. Scheduling Policies 3 2 5 Shortest first Longest first 5 3 3 5 2 2

4. The Game Nash equilibrium (NE): no user wants to change his machine NE can be far from optimal Cost (NE) = 3 Cost (OPT) = 1 PoA = Cost (Worst NE)/Cost (OPT) PoA = 3/1 PoA can be unbounded  Job 1 3 Job 2 Job 3 Job 1 1 Job 1 Job 2 Job 3 M1 M2 M3 1 3 3 Job1 3 1 3 Job2 3 3 1 Job3 Longest first

5. Coordination Mechanisms Clever scheduling policies Examples: Longest first Shortest first Can be preemptive t1 Job2 t2 t1 Job2 Job1 Job1 t2 Job2 t1 Job1 Job1 Job1 t2 Job2

6. Goal: Minimize PoA Shortest first Job 1 Job 2 Job 3 Job 1 Job 1 Job 2 Job 3 M1 M2 M3 1 3 3 Job1 3 1 3 Job2 3 3 1 Job3

7. History of Coordination Mechanisms Introduced by Christodoulou et. al in 2004

8. Open Problem Can we achieve Constant PoA using preemption or randomization? Azar, Jain, and Mirrokni (SODA 2008) Caragiannis (SODA 2009)

9. log m loglog m Our Results(1) • All deterministic mechanisms, even with preemption, if they are • symmetric • satisfy Independence of Irrelevant Alternatives (IIA) property have the PoA Ω ( ) .

10. log m loglog m Our Results(2) • All randomized mechanisms, even with preemption, if they are • symmetric • unbiased have the PoA Ω ( ) .

11. Symmetry a b t_x t_x All known mechanisms are symmetric. t_y t_y t_z t_z x x y y z z

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. t3(x) t2(y) t1(z) • 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 x y z

13. IIA Property Lemma: IIA each machine has order over the jobs - Proof is omitted. The order based on: • Jobs IDs (non-anonymous case) • Machines IDs (anonymous case)

14. Anonymous Case M4 M1 M2 M3 M4 Job Job Job

15. log m loglog m k-1 k 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 m = 1+k + k(k-1) + …… + k! k = Θ( )

16. log m loglog m log m loglog m Lower Bound for Non-Anonymous Case Cost (OPT) = 1 Cost (NE) >= k PoA >= k k = Ω ( ) PoA = Ω ( )

17. w y x z w x e g q w j n o p q k l m f x a y y x z x z y Lower Bound for Non-Anonymous Case w x y z

18. Lower Bound for Non-Anonymous Case a b t_x z t_x x b a t_y t_z y x y z

19. Lower Bound for Anonymous Case m3 m2 m1 m2 m3 m1

20. Lower Bound for Anonymous Case

21. Lower Bound for Anonymous Case a b t_x z t_x x b a t_y t_z y x y z

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. log m loglog m 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.

24. Conclusion • Achieving constant PoA using preemption or randomization is impossible. • If the inefficiency is bounded then we can achieve constant PoA.