VIEWS: 11 PAGES: 30 POSTED ON: 10/9/2011
Efficient coordination mechanisms for selfish scheduling Ioannis Caragiannis University of Patras & RACTI 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 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 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) 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 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 Coordination mechanisms: examples P(i,Nj) ShortestFirst/LongestFirst: 8 Order the jobs assigned to the same machine in 11 non-decreasing/non-increasing order of 5 processing times 47 3 Break ties according to the job IDs 4 1 Makespan: 19 0 Process the jobs assigned to the same machine 4 11 in parallel so they all complete in time equal to the machine load 8 Randomized: 47 Process the jobs non-preemptively in random 4 4 order 11 3 19 11 0 j An example: Makespan An example: ShortestFirst An example: LongestFirst 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 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 An example: PoA of Makespan and LongestFirst 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 Examples of potential functions Makespan Sort the load vector lexicographically ShortestFirst Sort the job completion times lexicographically LongestFirst and Randomized No potential function 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 Some results CM PoA Pot. PNE IDs Characteristics ShortestFirst Θ(m) Yes Yes Yes Strongly local, non-preemptive LongestFirst unbounded No No Yes Strongly local, non-preemptive Makespan unbounded Yes Yes No Strongly local, preemptive Randomized Θ(m) No ? No Strongly local, non-preemptive AJM-1 O(logm) No No Yes Local, non-preemptive AJM-2 O(log2m) Yes Yes Yes Local, preemptive, uses m ACOORD Θ(logm) Yes Yes Yes Local, preemptive, uses m O(mε ) Local, Preemptive BCOORD O(logm/loglogm) No ? No Local, preemptive, uses m O(m1/2) Yes Yes No Local, preemptive CCOORD O(log2m) Yes Yes No Local, preemptive, uses m O(mε ) Local, preemptive 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 Makespan vs. the ℓp norm of the machine loads ℓp-norm of the machine loads 1/p p L(N) p L(N j ) j Makespan is the ℓ∞-norm max L(Nj ) L(N) p m1/p max L(Nj ) j j 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 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 ACOORD definition P(i,Nj)=(ρij)1/pL(Nij) The schedule is always feasible ACOORD analysis (1) For each PNE N and optimal assignment O: max P(i,Nj ) L(N) p max L(O j ) j,iN j j 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 max P(i,N j ) P(i*, Nj1 ) P(i*, N j2 w i* j2 ) j,iN j L(N ij* ) min w i* j L(N) p max L(O) 2 j j ACOORD analysis (2) For each PNE N and optimal assignment O: max P(i,Nj ) (e(p 1)m1/(p 1) 1) max L(O j ) j,iN j j 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 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 BCOORD analysis For each PNE N and optimal assignment O: max P(i,Nj ) L(N) p max L(O j ) j,iN j j For each PNE N and optimal assignment O: 2p 1 1/(p 1) max P(i, Nj ) ln(p 1) m 1 max L(Oj ) j j,iNj PoA is at most O(logm/loglogm) when p= Θ(logm) and O(mε) when p=1/ε-1 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 k k! Ψk (A {b}) b tΨk t (A) t 0 (k t)! CCOORD definition: P(i,Nj)=(ρijΨp(Nj))1/p 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) Price of anarchy/stability Let N and O be two assignments such that Φ(N) ≤ cp+1 Φ(O). Then, max P(i,Nj ) c(p 1)m1/(p 1) 1 max L(O j ) j,iN j j 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 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?