Truthful Algorithms for Scheduling Selfish Tasks on Parallel Machines

1
Truthful Algorithms for Scheduling Selfish Tasks
on Parallel Machines
WINE 2005

• Eric Angel, Evripidis Bampis, Fanny Pascual
• LaMI, University of Evry, France

2
Outline

• Introduction
• Truthful algorithm
• Truthful coordination mechanism
• Conclusion

3
Introduction

• Aim To optimize the performances of a network
  used by selfish agents.
• A scheduling problem
• Koutsoupias, Papadimitriou STACS99
• Cj completion time of task j. (e.g. C32)

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 ICALP04

5
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
• .

6
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 ?

7
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

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

• 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?

• SPT?
• Schedules tasks 1,2,,n s.t. l1 lt l2 lt lt ln
• Task (i1) starts when 1/m of task i has been
  executed.
• Example (m3)

1
7
4
2
5
8
3
6
9
12
Algorithm SPT?

• Thm SPT? is (2-1/m)-approximate.
• Idea of the proof (m3)
• Idle times
• idle_beginning(i) ? (1/3 lj)
• idle_middle(i) 1/3 ( li-3 li-2 li-1 )
  li-3
• idle_end(i) li1 2/3 li idle_end(i1)

jlti
13
Algorithm SPT?

• Thm SPT? is (2-1/m)-approximate.
• Idea of the proof (m3)

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/(m1), returns SPT?.
• With a proba. 1/(m1), returns LPT.
• The expected approx. ratio of SPT? - LPT is
  smaller than the one of SPT e.g. for m2,
  ratio(SPT?-LPT) lt 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 bgtli. It is now larger
  than tasks 1,, x, smaller than task x1.
• l1 lt lt li lt li1 lt lt lx lt lx1 lt
  lt ln
• LPT decrease of Ci(lpt) (li1 lx)
• SPT? increase of Ci(spt?) 1/m (li1 lx)
• SPT?-LPT
• change - m/(m1) Ci(spt?) 1/(m1) Ci(spt?)
  0

b lt
16
Outline

• Introduction
• Truthful algorithm
• Truthful coordination mechanism (m2)
• Coordination mechanism definition
• A truthful coordination mechanism
• Conclusion

17
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
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 pgt2/3, SM(p) is truthful if the tasks
  are powers of C (4-3p)/(2-p).
• Thm 2 When plt1/2, SM(p) is not truthful even if
  the tasks are powers of any integer Cgt1.

20
A truthful coordination mechanism

• Thm 2 When plt1/2, SM(p) is not truthful even if
  the tasks are powers of any integer Cgt1.
• 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