Efficient coordination mechanisms for unrelated machine scheduling by hcj

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									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,iN 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,iN 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,iN 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,iN 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,iNj
                                             
   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,iN 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?

								
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