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

Outline  Introduction  Truthful algorithm  Truthful coordination mechanism  Conclusion

Introduction Aim: To optimize the performances of a network used by selfish agents. A scheduling problem: [Koutsoupias, Papadimitriou: STACS’99] C j = completion time of task j. (e.g. C 3 =2) time 0123 n tasks m machines

Introduction Game theoretic approach: –Task i has a secret real length l i. –Each task bids a value b i ≥ l i. –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]

Introduction Each task wish to reduce its completion time (and may lie if necessarily). 2 models: –Model 1: If i bids b i, its length is l i –Model 2: If i bids b i, its length is b i Example: We have 3 tasks:,, Task 1 bids 2.5 instead of 1:. 123 Model 1: C 1 = 1 Model 2: C 1 = 2.5 time

SPT: a truthful algorithm 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 ? 1 2 3

LPT 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 : 123 Task 1 has incentive to bid 2.5, and LPT is not truthful. C 1 = C 1 = 1 time

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.

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

SPT-LPT is not truthful 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 : SPT :LPT : C 1 = p + 3(1-p) = 3 - 2p 1 23 SPT : LPT : C 1 = 1 1 1

Algorithm SPT  SPT  : Schedules tasks 1,2,…,n s.t. l 1 < l 2 < … < l n Task (i+1) starts when 1/m of task i has been executed. Example : (m=3)

Algorithm SPT  Thm: SPT  is (2-1/m)-approximate. Idea of the proof: (m=3) Idle times : idle_beginning(i) = ∑ (1/3 l j ) idle_middle(i) = 1/3 ( l i-3 + l i-2 + l i-1 ) – l i-3 idle_end(i) = l i+1 – 2/3 l i + idle_end(i+1) j<i

Algorithm SPT  Thm: SPT  is (2-1/m)-approximate. Idea of the proof: (m=3) Cmax = (∑(idle times) + ∑(li)) / m ∑(idle times) ≤ (m-1) l n and l n ≤ OPT  Cmax ≤ ( 2 – 1/m ) OPT Cmax

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.

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

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

Coordination mechanism 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 :,, M SPT M LPT Denoted by Mixt

A truthful coordination mechanism SM(p): p SPT – (1-p)Mixt –M 1 schedules tasks with SPT. –M 2 schedules tasks with SPT with a proba. p, and with LPT otherwise. Expected approx. ratio = 4/3 + p/6.

We consider the 2 nd 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. A truthful coordination mechanism

Thm 2: When p 1. Idea of the proof: Policy of M1 = SPT C2C2 C M2M2 M1M1 C C2C2 … … xC/2 tasks C CCC2C2 C2C2 … … x/2 tasks Policy of M2 = SPT C2C2 C M2M2 M1M1 CC C2C2 C2C2 CC … … C x C tasks x tasks Policy of M2 = LPT C3C3 C3C3 C i increases of:  spt = C 3 - C + (x/2)C 2 C i decreases of:  mixt =xC 2 + C - C 3 Overall change = p  spt - (1-p)  mixt

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