Truthful Approximation Mechanisms for Scheduling Selfish Related Machines

1. Truthful Approximation Mechanisms for Scheduling Selfish Related Machines Motti Sorani, Nir Andelman & Yossi Azar Tel-Aviv University

2. The Classic Scheduling Problem • Scheduling jobs on uniformly related machines • n Jobs: • m machines with different speeds: • The objective: minimize the maximum completion time over all machines (Makespan) • Known to be NP-Complete • Several approximation-algorithms

3. Scheduling Jobs on Selfish Related Machines • One-Parameter problem • Machines are owned by rational selfish agents

4. Scheduling Jobs on Selfish Related Machines • The machine’s speed is known to its owner only.Thesecret: the cost per unit of work

5. Scheduling Jobs on Selfish Related Machines • Job sizes are common knowledge • The system wants to execute the jobs while minimizing the makespan

6. Scheduling Jobs on Selfish Related Machines • Bidding

7. Scheduling Jobs on Selfish Related Machines • The cost of machine is where is the total amount of work assigned to it. • The machines getpaid. The goal of each machine is to maximize its profit.

8. Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3

9. Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3 First Phase: Bidding

10. Watching the Game The jobs: 12,10, 7, 5, 4, 4, 3 Second Phase: The system allocates the jobs to the machines according to their declared bids, and simultaneously delivers the payments Makespan=12 3 5 10 4 7 12 4 16 15 13

11. Truthful Mechanism Design M=(A,P) A : Allocation Algorithm P : Payment

12. Truthful Mechanism Design M=(A,P) A : Allocation Algorithm P : Payment

13. Mechanism Design • The idea: Overcome selfishness by payments • Mechanism M=(A,P) • Strategy for agent : • The outcome of the algorithm is • The work allocated to agent : • The payment to agent : • The profit of agent : Observation: Paying each agent its cost is not truthful

14. Truthful Mechanism • Dominant strategy for agent : • Truthfulness: truth-telling ( ) is a dominant strategy for each agent • VCG is not applicable as the objective is not utilitarian (maximize “social welfare”) Our goal: Design a Truthful Mechanism M=(A,P)which approximates the Minimum Makespan

15. Truthful Mechanism • Consider the work assigned to agent as a single-variable function of • Work-curve Truthfulness <=> Monotone Algorithm

16. Truthful Mechanism Theorem [Archer and Tardos]: A mechanism is truthful and admits a voluntaryparticipation iff (a) the work-curve for each agent is decreasing,(b) and the payments in this case should be

17. Monotone Algorithms • Truthful Mechanism: • Monotone Algorithm • Payment scheme • The work-curve profit cost

18. Overbidding less profit lesspayment slower faster

19. Underbidding loss

20. Previous Results - Approximation • Gonzalez et al: 2-approximation LPT greedy assignment • Horowitz and Sahni: FPTAS for constant number of machines • Hochbaum and Shmoys:PTAS for arbitrary number of machines All these algorithms are not monotone

21. Previous Results – Mechanism Design • Monotone Algorithm (not polytime) [Archer & Tardos]: • optimal solution • satisfies voluntary participation Among the optimal allocations of jobs, select the one in which the work-vector is lexicographically minimum.

22. Previous Results – Mechanism Design Scenario: gradual slowdown Slowing down

23. Previous Results – Mechanism Design • Classic approximation algorithms are not monotone. • Archer and Tardos: randomized truthful 3-approximation mechanism (truthful in expectation) • Auletta et al: deterministic truthful (4+ε)-approximation mechanism for any fixed number of machines

24. Notions of Truthfulness • Truthfulness in expectation: bidding truthfully always maximizes the agent's expected profit • Universal truthfulnessbidding truthfully always maximizes an agent's profit, no matter what the other agents bid, and no matter what are the outcomes of the mechanism's random coin flips

25. Our Results • Deterministic 5-approximation truthful mechanism for arbitrary number of machines • Deterministic truthful (F)PTAS for any fixed number of machines We now show a simplified version for arbitrary number of machines which achieves a 12-approximation truthful mechanism.

26. Valid Fractional Assignment • Given a threshold T, treat the machines as bins of size T/bi • Fractional Assignment – Partition each job to pieces, assign the pieces to the bins • Valid Fractional Assignment • Each bin is large enough to contain all pieces assigned to it • For every piece assigned to a bin, the bin is capable of containing the entire job (which the piece belongs to) • – The smallest threshold for which a valid fractional assignment exists.

27. Valid Fractional Assignment • Example • Jobs: 7,5,4,3,3,2 • Bids: 1/5, 1/4, 1/3 • Threshold = 2 3 2 5 4 3 7 5 3 bin size: 6 8 10

28. Valid Fractional Assignment • can be calculated in a greedy manner • is a lower-bound to Opt

29. Monotonicity of Tf • Observation: • behaves in a “monotone manner” • For any machine i which is not the fastest (i>1)

30. Truthful Mechanism for arbitrary number of Machines • Guidelines of algorithm Monotone-RF • Round the bids to the closest power of 4 • Sort the jobs in non-increasing order • Calculate a valid fractional assignment and an appropriate threshold Tf • Assign jobs (using the rounded bids) in non-increasing order of size, from the fastest to the slowest (breaking ties by external ID) • The first machine – until a threshold of 2Tf is exceeded • Rest of the machines – until a threshold Tf is exceeded • Return the assignment Init: fractional: rounding:

31. Truthful Mechanism for arbitrary number of Machines slowest fastest …

32. Monotonicity of Monotone-RF • Intuition: Assigning jobs according the rounded bids forces non-increasing work-curves • From now on we assume the bids are equal to the real speeds. • We shall show : Slowing down => Less/Equal Amount of work

33. Why Rounding the Bids Helps? • behaves in a “monotone manner” • For any machine i which is not the fastest (i>1) • Say the rounded bid is multiplied by 4 Total work is At most Total work is At least

34. Scenarios of Slowdowns • The unique fastest behaves differently:Rounding is not enough • The bad scenario: The fastest machine slows down one step (the rounded bid is multiplied by 4) and some other machine becomes the fastest

35. Scenarios of Slowdowns • Solution: Double the threshold for the bin of the unique fastest machine • When it slows down one step ,two cases: • Remains the unique fastest: • Bin size can not increase • Jobs are allocated in the same order • No longer the fastest: losing the doubled threshold balances the possible increase of the threshold.

36. Analysis for Partially-Full and Empty Machines • So far we considered full machines only The Red Machine slows down

37. Monotonicity by Gradual Slowdown • Monotone-RF is monotone. Hence a Mechanism based on Monotone-RF and payment schemeis truthful

38. Truthfulness - Remark • The work-curve

39. Approximation Analysis • The first bin capacity: • is a lower-bound to Opt. • Speeds were rounded to powers of 4 • A Total of 12-approximation

40. Guidelines for 5-Approximation • Prefer the fastest machine already in the rounding phase. • Make sure the first bin is at least 4 times the second one • For any machine i which is not the fastest (i>1) • Speeds can be rounded to powers of 2.5 • A Total of 5-approximation

41. Truthful PTAS-Mechanism for Any Fixed Number of Machines Exact Minimum-Lexicographically Solution

42. Truthfulness of Monotone-PTAS • Job sizes were generated independently from the bids • The optimal Min. Lex. Solution is monotone

43. Approximation Analysis • Running Time is linear • Assume we do not use machines slower than • The chunks add multiplicative overhead of (1+ε) • The assumption above adds another multiplicative overhead of (1+ε)

44. Guidelines for the FPTAS • Uses any c-approximation algorithm as a black box, generates a c(1+ε)-approximation • Rounds the bids to powers of (1+ε) • Calculate all possible (sorted) assignments made by the black box • Try all assignments on the given rounded bids-vector. Pick the one with minimal makespan, or if more than one exists, the one which is lexicographically maximum.

45. Conclusions and Open Problems • We have shown • Deterministic 5-approximation truthful mechanism for assigning jobs on related machines • A (F)PTAS truthful mechanism for any fixed number of machines • Is there a PTAS truthful mechanism for arbitrary number of Machines?