1. On Designing Truthful Mechanisms for Online Scheduling V. Auletta, R. De Prisco, P.P. and G. Persiano Università di Salerno

2. The Internet Open, self organized, no central authority, anarchic Different components which have their own goal may not follow the protocol Selfish agents

3. The Internet Open, self organized, no central authority, anarchic An Autonomous System may report false link status to redirect traffic to another AS AS1 AS2 source destination Link down

4. Routing/Scheduling Unsplittable traffic J 1, J 2,…,J n We look at the network congestion (makespan) sourcedestination Scheduling Selfish Machines: Selfish users own the links and privately know their speeds s1s1 smsm s2s2 0 0 0 W i = J k assigned to machine i max i W i /s i

5. Mechanism design Mechanism: M=(A,P) Computes a solution X=A(r 1,r 2,…, r i,…,r m ) Provides a payment P i (r 1,r 2,…, r i,…,r m ) Agents GOAL: maximize their own utility u i (r i ) := P i (r 1,r 2,…, r i,…,r m ) – cost i (X,s i ) cost i (X,s i ) = w i /s i s 1,s 2,…, s i,…,s n true input

6. Mechanism design Strategyproof mechanisms: no incentive to lie (report r i s i ) u i (s i ) u i (r i ) (truth-telling is the best strategy)

7. 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]

8. Example: Greedy Algorithm 1 1+ 2 2 (1+ ) 2 1+ 1 3 2 2 3 NOT MONOTONE

9. 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]

10. Monotonization techniques A mon A M=(A mon,P) AlgorithmMechanism M=(A,P)A hard loss of performance

11. Our contribution (1/2) A black-box, polytime A mon A easy c-apx c = c(1+ ) c < c c Offline: Online: Jobs arrive one by one, no reallocation! must loose something A mon A hard c-comp (the case of two machines) Proved for any m = O(1) in [Andelman & Azar & Sorani, STACS05]

12. 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.

13. 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 1 1 1 Proof: 1 r 1 r r/opt = r/1=r r r 1+1/r 1

14. Upper Bound (m = 2) Theorem 5: A mon A c-comp c max {cr, 1+1/r}, r 1 c is the comp. ratio on identical speeds Corollary 5: Greedy mon Greedy 3/2-comp c-comp c= 1.823... Lower bound: =1.62… (2 jobs). (1-comp) ( -comp)

15. 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

16. w1w1 w2w2 … wiwi … wmwm sisi s1s1 s2s2 smsm w 1 w 2 … w i … w m s i s1s1 s2s2 smsm w i > 0 s i > s i w i > 0 Weakly Monotone algorithms: Mechanisms With Verification

17. w1w1 w2w2 wiwi wmwm s i s1s1 s2s2 s m An 8-competitive algorithm: 2opt … … JkJk JkJk JkJk JkJk JkJk JkJk Try UB = 1, 2, 4, 8,... stop UB 2opt UB

18. Mechanisms With Verification s1s1 smsm sisi Problem: JkJk hole s j >s i >s i+1 s j+1 machine shifts sjsj JkJk no work

19. Mechanisms With Verification Fix: Avoid Holes JkJk JkJk JkJk JkJk OK NO, Reallocate JkJk JkJk

20. Mechanisms With Verification Fix: Analysis JkJk Original alg Reallocated 8opt 4opt 12-comp. mechanism

21. Open Questions Close the gaps: 2 1.62… 1.823… O(1) 1.62… ???? m Lower Bound Upper Bound No verification any 1.62… 12 Verification

22. Thank You