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

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

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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

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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

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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

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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)

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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]

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

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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]

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

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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]

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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.

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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

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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)

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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

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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

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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

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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

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

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

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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

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Thank You

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