Efficient coordination mechanisms for selfish scheduling

1. Efficient coordination mechanisms for selfish scheduling Ioannis Caragiannis University of Patras & RACTI

2. What is this talk about? • Design (or redesign) the game so that the price of anarchy is minimized • Approaches: • Taxes or tolls in network routing • Stackelberg routing/scheduling strategies • Protocol design in network and cost allocation games • Coordination mechanisms

3. Unrelated machine scheduling • m machines • n jobs each having a load vector • wij is the processing time of job i when it is processed by machine j

4. Unrelated machine scheduling • m machines • n jobs each having a load vector • wij is the processing time of job i when it is processed by machine j • Objective: • To assign each job to a machine so that the maximum completion time among all jobs is minimized • Equivalently, the maximum (makespan) of the machine loads is minimized • Well understood problem in terms of its offline and online approximability • Lenstra, Shmoys, & Tardos (Math. Programming 1990) • Aspnes, Azar, Fiat, Plotkin, & Waarts (JACM 1997) • Azar, Naor, & Rom (J. Algorithms 1995)

5. Selfish scheduling • The setting: • Each job is owned by a selfish agent that aims to minimize the completion time of her job • Coordination mechanism (CM): • A scheduling policy within each machine • Defines a game among the jobs • Our goal: • To design CMs that guarantee that the assignments reached are efficient

6. Games induced by coordination mechanisms • The jobs are the players • Each job has all machines as strategies • Assignment N: one strategy per player • Nj denotes the set of jobs assigned to machine j • L(Nj) denotes the load of machine j • Scheduling policy: • Defined by the completion time P(i,Nj) of job i when it is assigned to machine j • It should always produce feasible schedules! • Induced game: • The cost of each player is her completion time

7. Coordination mechanisms: examples P(i,Nj) 8 • ShortestFirst/LongestFirst: • Order the jobs assigned to the same machine in non-decreasing/non-increasing order of processing times • Break ties according to the job IDs • Makespan: • Process the jobs assigned to the same machine in parallel so they all complete in time equal to the machine load • Randomized: • Process the jobs non-preemptively in random order 11 5 47 3 4 1 19 0 11 4 8 47 4 4 3 11 19 11 0 j

8. An example: Makespan

9. An example: ShortestFirst

10. An example: LongestFirst

11. Coordination mechanisms: characteristics • Non-preemptive • Process jobs uninterruptedly according to some order • Preemptive • May interrupt jobs and introduce idle times (delay) • Strongly local • The only information required in order to compute the schedule within a machine is the processing times of the jobs assigned to the machine • Local • May use the whole load vector of the jobs assigned to the machine • Anonymous jobs • When no ID information is associated to the jobs

12. Efficiency measures • Pure Nash Equilibria (PNE): • Assignments from which no player has an incentive to unilaterally deviate • Price of anarchy (PoA)/stability (PoS): • The maximum/minimum among all PNE of the ratio of the maximum completion time over the optimal makespan • Approximation ratio of a CM: • The maximum of the PoA of the induced game over all input instances • Our goals: • Small approximation ratio • PNE should exist and should be easy to find

13. An example: PoA of Makespan and LongestFirst

14. Potential games • Definition: • A potential function can be defined on the assignments so that for any two assignments differing only in the strategy of a player, the difference on the values of the potential and the difference of the player’s cost have the same sign • Implies that: • The Nash dynamics is acyclic • The state with minimum potential is PNE • A desired property: • Convergence to PNE after a polynomial number of selfish (usually best-response) moves

15. Examples of potential functions • Makespan • Sort the load vector lexicographically • ShortestFirst • Sort the job completion times lexicographically • LongestFirst and Randomized • No potential function

16. Related work • Work directly related to CMs • Christodoulou, Koutsoupias, & Nanavati (ICALP ’04/TCS) • Immorlica, Li, Mirrokni, & Schultz (WINE ’05/TCS) • Results about ShortestFirst, LongestFirst, Randomized, and Makespan in several machine models • Azar, Jain, & Mirrokni (SODA ’08) • Limitations of (strongly) local non-preemptive CMs • Two CMs (henceforth called AJM-1 and AJM-2) that use the notion of the job inefficiency ρij = wij/wi,min • C (SODA ’09) • Fleischer & Svitkina (ANALCO ’10) • Limitations of local non-preemptive CMs

17. Some results

18. Main ideas • Scheduling policies: • Preemptive (with idle times) • Local, job completion times depend on inefficiency • Defined using an integer parameter p • set to O(logm) in order to obtain our best results • or set to a large constant

19. Makespan vs. the ℓpnorm of the machine loads • ℓp-norm of the machine loads • Makespan is the ℓ∞-norm

20. ACOORD ideas • General idea: • use the job IDs so that the scheduling policy simulates an online algorithm for minimizing the makespan • How? By defining P(i,Nj) in terms of jobs with the i smallest IDs • Ni: the restriction of assignment N to the jobs with the i smallest IDs • Example: P(i,Nj)=L(Nij) simulates a simple greedy online algorithm known to be at least Ω(m)-approximate • Aspnes, Azar, Fiat, Plotkin, & Waarts (JACM 1997) • Convergence to PNE in at most n adversarial rounds of best-response moves

21. ACOORD ideas (contd.) • Better online algorithms, e.g., the greedy algorithm for the ℓp-norm • Awerbuch, Azar, Grove, Kao, Krishnan, & Vitter (FOCS ’95) • C (SODA ’08) • For p=O(logm), it gives O(logm)-approximation to the makespan • Unfortunately, the online criterion does not seem to translate always to feasible schedules

22. ACOORD definition • P(i,Nj)=(ρij)1/pL(Nij) • The schedule is always feasible

23. ACOORD analysis (1) • For each PNE N and optimal assignment O: • Proof: Let i* be a job with maximum completion time that it is assigned to machine j1 in N and has inefficiency 1 in machine j2

24. ACOORD analysis (2) • For each PNE N and optimal assignment O: • Proof sketch: relate the ℓp+1-norms of the machine loads using the following argument: • In N, why doesn’t job i use the machine it uses in O? • Use of convexity properties of polynomials, Minkowski inequalities, etc. • PoA is at most Θ(logm) when p= Θ(logm) and O(mε) when p=1/ε-1

25. BCOORD • P(i,Nj)=(ρij)1/pL(Nj) • The schedule is always feasible • Anonymous jobs • Unfortunately, the existence of PNE is not guaranteed by potential function arguments • The Nash dynamics may contain cycles • Simple examples with 4 machines and 5 basic jobs

26. BCOORD analysis • For each PNE N and optimal assignment O: • For each PNE N and optimal assignment O: • PoA is at most O(logm/loglogm) when p= Θ(logm) and O(mε) when p=1/ε-1

27. CCOORD • For integer k ≥ 0, Ψk is defined as Ψk(Ø)=0, Ψ0(A)=1, and Ψk(A) is the sum of all monomials with elements of A of total degree k multiplied by k! • E.g., • Ψ2({a,b})=2(a2+b2+ab) • Ψ3({a,b,c})=6(a3+b3+c3+a2b+ab2+a2c+ac2+b2c+bc2+abc) • Some properties: • L(A)k ≤ Ψk(A) ≤ k! L(A)k • CCOORD definition: P(i,Nj)=(ρijΨp(Nj))1/p

28. A potential function • The function Φ(N)=ΣjΨp+1(Nj) is a potential function for the game induced by CCOORD • Actually, for any two assignments N and N’ differing only in the strategy of player i, it holds that • Φ(N) - Φ(N’) = (p+1)wi,min(P(i,Nj1)p - P(i,N’j2)p) • i.e, the game is an exact potential game and, hence, equivalent to a congestion game • Monderer and Shapley (GEB 1996)

29. Price of anarchy/stability • Let N and O be two assignments such that Φ(N) ≤ cp+1 Φ(O). Then, • By considering a PNE N with minimum potential and an optimal assignment O (i.e., c ≤ 1): • PoS = O(logm) when p = Θ(logm) • For any PNE, it is c ≤ (p+1)/ln2 • PoA is at most O(log2m) when p = Θ(logm) and O(mε) when p=1/ε-1

30. Open problems • Constant approximation ratio? • Is the case of anonymous jobs provably more difficult? • Is there a non-preemptive local CM that induces potential games and has approx. ratio o(m)? • Does the game induced by BCOORD have PNE? • What is the complexity of computing PNE in the game induced by CCOORD? • Even if PNE are hard to find, does the game induced by CCOORD converge to efficient assignments after a polynomial number of adversarial rounds? • Mixed Nash Equilibria? Other equilibria?