Truthful Algorithms for Scheduling Selfish Tasks on

Parallel Machines

Eric Angel, Evripidis Bampis, Fanny Pascual

LaMI, University of Evry, France

WINE 2005

OutlineIntroduction

Truthful algorithm

Truthful coordination mechanism

Conclusion

IntroductionAim: 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)

2

13

1 21 3

time0 1 2 3

n tasksm machines

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]

Introduction• 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:

.

1 2 3

Model 1: C1 = 1Model 2: C1 = 2.5

time

31 2

0 1 2 3 4 5

1

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:

1 2 3

Task 1 has incentive to bid 2.5, and LPT is not truthful.

C1 = 332 1

C1 = 1

time

31 2

0 1 2 3 4 5time0 1 2 3 4 5

1

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 algorithmSPT-LPT is not truthfulAlgorithm: SPTA 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 : 2.5

1 2 3

12

3 32 1SPT : LPT :

C1 = p + 3(1-p) = 3 - 2p

12 3SPT :

LPT : 31 2

C1 = 1 1 1

Algorithm SPT• 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)

0 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

5

6

7

8

9

4

Algorithm SPT• 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-3idle_end(i) = li+1 – 2/3 li + idle_end(i+1)

j<i

0 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

5

6

7

8

9

4

Algorithm SPT• Thm: SPT is (2-1/m)-approximate.• Idea of the proof: (m=3)

0 1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

5

6

7

8

9

4

Cmax = (∑(idle times) + ∑(li)) / m ∑(idle times) ≤ (m-1) ln and ln ≤ 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>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 <

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 : , ,1 2 3

1 2

3

MSPT

MLPT

Denoted by Mixt

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.

• 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.

A truthful coordination mechanism

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

C2C

M2

M1 C

C2……

xC/2 tasks

C C C C2

C2……

x/2 tasks

Policy of M2 = SPT

C2

C

M2

M1 C C

C2 C2

CC

…… C

x C tasks

x tasks

Policy of M2 = LPT

C3

C3

Ci increases of: spt = C3 - C + (x/2)C2 Ci decreases of: mixt =xC2 + C - C3

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