Truthful Algorithms for Scheduling Selfish Tasks on Parallel Machines

1. WINE 2005 Truthful Algorithms for Scheduling Selfish Tasks on Parallel Machines Eric Angel, Evripidis Bampis, Fanny Pascual LaMI, University of Evry, France

2. Outline • Introduction • Truthful algorithm • Truthful coordination mechanism • Conclusion

3. Introduction 1 n tasks m machines 2 2 3 1 1 3 0 1 2 3 time • Aim: To optimize the performances of a network used by selfish agents. • A scheduling problem: • [Koutsoupias, Papadimitriou: STACS’99] • Cj = completion time of task j. (e.g. C3=2)

4. Introduction • Game theoretic approach: • Task i has a secret real length li. • Each task bids a value bi ≥ li. • Each task knows the values bidded by the other tasks, and the algorithm. • Each task wish to reduce its completion time. • Social cost = maximum completion time (makespan) • Aim : An algorithm truthful and which minimizes the makespan. [Christodoulou, Koutsoupias, Nanavati: ICALP’04]

5. Introduction 1 2 3 Model 1: C1 = 1 Model 2: C1 = 2.5 3 1 2 1 time 0 1 2 3 5 4 • Each task wish to reduce its completion time (and may lie if necessarily). • 2 models: • Model 1: If i bids bi, its length is li • Model 2: If i bids bi, its length is bi • Example: We have 3 tasks: , , Task 1 bids 2.5 instead of 1: .

6. SPT: a truthful algorithm 1 3 2 • SPT: Schedules greedily the tasks from the smallest one to the largest one. • Example: • Approx. Ratio = 2 – 1/m [Graham] • Are there better truthful algorithms ?

7. LPT time 0 1 2 3 5 4 1 2 3 C1 = 3 3 1 2 C1 = 1 3 1 2 1 time 0 1 2 3 5 4 • LPT: Schedules greedily the tasks from the largest one to the smallest one. • Approx. Ratio = 4/3 – 1/(3m) [Graham] • We have 3 tasks: , , Task 1 bids 1: Task 1 bids 2.5: Task 1 has incentive to bid 2.5, and LPT is not truthful.

8. Introduction • Idea: to combine: • A truthful algorithm • An algorithm not truthful but with a good approx. ratio. • Task: wants to minimizes its expected completion time. • Our Goal: A truthful randomized algorithm with a good approx. ratio.

9. Outline • Introduction • Truthful algorithm • SPT-LPT is not truthful • Algorithm: SPT • A truthful algorithm: SPT-LPT • Truthful coordination mechanism • Conclusion

10. SPT-LPT is not truthful 1 3 2 3 1 2 3 1 2 1 2 3 1 1 • Algorithm SPT-LPT: • The tasks bid their values • With a proba. p, returns an SPT schedule. With a proba. (1-p), returns an LPT schedule. • We have 3 tasks : , , • Task 1 bids its true value : 1 • Task 1 bids a false value : 2.5 1 2 3 C1 = p + 3(1-p) = 3 - 2p SPT : LPT : LPT : C1 = 1 SPT :

11. Algorithm SPT 12 0 1 2 3 4 5 6 7 8 9 11 10 • SPT: Schedules tasks 1,2,…,n s.t. l1< l2< … < ln Task (i+1) starts when 1/m of task i has been executed. • Example: (m=3) 1 4 7 2 8 5 3 6 9

12. Algorithm SPT 1 4 7 2 5 8 3 6 9 12 0 1 2 3 4 5 6 7 8 9 11 10 • Thm: SPT is (2-1/m)-approximate. • Idea of the proof: (m=3) Idle times : idle_beginning(i) = ∑ (1/3 lj) idle_middle(i) = 1/3 ( li-3 + li-2 + li-1 ) – li-3 idle_end(i) = li+1 – 2/3 li + idle_end(i+1) j<i

13. 1 4 7 2 5 8 3 6 9 12 0 1 2 3 4 5 6 7 8 9 11 10 Algorithm SPT • Thm: SPT is (2-1/m)-approximate. • Idea of the proof: (m=3) Cmax Cmax = (∑(idle times) + ∑(li)) / m ∑(idle times) ≤ (m-1) ln and ln≤ OPT Cmax ≤ ( 2 – 1/m ) OPT

14. A truthful algorithm: SPT-LPT • Algorithm SPT-LPT: • With a proba. m/(m+1), returns SPT. • With a proba. 1/(m+1), returns LPT. • The expected approx. ratio of SPT - LPT is smaller than the one of SPT: e.g. for m=2, ratio(SPT-LPT) < 1.39, ratio(SPT)=1.5 • Thm: SPT-LPT is truthful.

15. A truthful algorithm: SPT-LPT • Thm: SPT-LPT is truthful. • Idea of the proof: • Suppose that task i bids b>li. It is now larger than tasks 1,…, x, smaller than task x+1. • l1 < … < li < li+1 < … < lx < lx+1 < … < ln • LPT: decrease of Ci(lpt) ≤(li+1 + … + lx) • SPT: increase of Ci(spt) = 1/m (li+1 + … + lx) • SPT-LPT: • change = - m/(m+1) Ci(spt) + 1/(m+1) Ci(spt)≥ 0 b <

16. Outline • Introduction • Truthful algorithm • Truthful coordination mechanism (m=2) • Coordination mechanism: definition • A truthful coordination mechanism • Conclusion

17. Coordination mechanism 3 1 2 MLPT MSPT • Coordination mechanism [CKN 04]: • Each machine has a local policy to schedule its tasks. This policy should not depend on the other tasks. • Each task chooses on which machine it will be scheduled. • Example : , , 1 2 3 Denoted by Mixt

18. A truthful coordination mechanism • SM(p): p SPT – (1-p)Mixt • M1 schedules tasks with SPT. • M2 schedules tasks with SPT with a proba. p, and with LPT otherwise. • Expected approx. ratio = 4/3 + p/6.

19. A truthful coordination mechanism • We consider the 2nd model: if i bids b, its execution time is b. • Thm 1: When p>2/3, SM(p) is truthful if the tasks are powers of C ≥ (4-3p)/(2-p). • Thm 2: When p<1/2, SM(p) is not truthful even if the tasks are powers of any integer C>1.

20. Policy of M2 = LPT Policy of M2 = SPT x C tasks … … C3 … M1 C C C C C C M1 C C C2 C2 … … … M2 C2 C2 C2 M2 C C C C2 C2 x tasks xC/2 tasks x/2 tasks C3 A truthful coordination mechanism • Thm 2: When p<1/2, SM(p) is not truthful even if the tasks are powers of any integer C>1. • Idea of the proof: Policy of M1 = SPT Ci increases of: spt = C3 - C + (x/2)C2 Ci decreases of: mixt =xC2 + C - C3 Overall change = p spt - (1-p) mixt

21. Conclusion • Conclusion: • A truthful centralized algorithm. • A truthful coordination mechanism under certain conditions. • Future work: • Better truthful coordination mechanisms • Case of uniform machines