Scheduling with Uncertain Processing Times€¦ · Stochastic Machine Scheduling with Precedence...
Transcript of Scheduling with Uncertain Processing Times€¦ · Stochastic Machine Scheduling with Precedence...
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This Talk
Mixture of
1. (mini) introductory lecture, stochastic scheduling
2. along with some recent results (unrelated machinescheduling) [Skutella, Sviridenko & U. 2016]
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1 Setting the Scene
2 Stochastic Scheduling
3 Unrelated Machines & Time-indexed LP
4 Final Remarks
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Single Machine Scheduling
Given: n jobs j with weights wj > 0, processing times pj ∈ Z>0
Task: sequence jobs on 1 machine; at most one job at a time;
0 timeCred Cgreen Corange Cblue
Objective: minimize∑
j wj Cj where Cj = j ’s completion time;
Theorem (Smith 1956)
Smith’s rule, sequencing jobs in order wj/pj is optimal
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Identical Parallel Machine Scheduling
Given: n jobs as above; m identical parallel machines
Task: schedule each job on any one machine; minimize∑
j wj Cj
0 time
Theorem
Problem is strongly NP-hard [Garey & Johnson, Problem SS13]Smith’s rule: tight 1.21-approximation [Kawaguchi & Kyan, 1986]There exists a PTAS [Skutella & Woeginger, 2000]
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Unrelated Machine Scheduling
Given: m machines, machine-dependent processing times pij
Task: schedule each job on one machine; minimize∑
j wj Cj
0 time
Theorem
Problem is APX-hard [Hoogeveen et al., 2002]
Exists ( 32 − c)-approximation [Bansal, Srinivasan, Svensson, 2016]
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Main Result
Theorem (Skutella, Sviridenko, U. 2016)
Stochastic unrelated machine scheduling has a( 3+∆
2 )-approximation.
∆ = bounds the (squared) coeff. of variation of processing times
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1 Setting the Scene
2 Stochastic Scheduling
3 Unrelated Machines & Time-indexed LP
4 Final Remarks
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Stochastic Scheduling
processing time = (independent) random variables Pj (or Pij); allknown to us
1
0time t
Pr[Pj ≥ t]
Solution: Non-anticipatory scheduling policy Π
Decisions based on information up to now and a priori knowledgeabout Pj (or Pij); no further information about the future.
0 timenow
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Optimality
On instance I with policy Π
Π(I ) := cost of policy Π on I , is a random variable
Definition (Optimal Policy)
Call ΠOPT optimal if it achieves
inf E[Π(I )] | Π non-anticipatory policy
Existence follows from [Mohring, Radermacher, Weiss 1985]
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An Example
n = 4 jobs, unit weights
time0 1 10
blue jobs: Pj = 1
green jobs: Pj =
0 probability 4/5
10 probability 1/5(note E[Pj ] = 2)
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Stochastic World
Tradeoff: better to delay
large E[Pj ] or large Pr(Pj“large”) (heavy tail) ?
Claim
Unique 2-machine optimal policy: green,blue → green → blue[with Π(I ) = E[
∑j Cj ] = 6.92].
Just work it out
0 1 2 1110
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Stochastic World: Deliberate Idleness
Theorem (U. 2003)
There are instances where only optimal policy deliberately leavesmachines idle.
0ε time
0ε time
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Approximation Algorithms
Optimal policies
intuitively complex, exponential size decision tree;definitely NP(APX)-hard, . . .
only computing E[Π(I )] can be #P-hard[Hagstrom, 1988]
Definition (Approximation)
Policy Π has performance guarantee α ≥ 1, if for all instances I
E[Π(I )] ≤ α E[ΠOPT(I )]
Our adversary is non-anticipatory, too!
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Approximation Algorithms
Mohring, Schulz & U. [JACM, 1999]First LP-based approximation algorithmse.g.: Smith’s rule has performance guarantee ( 3+∆
2 ).
Skutella & U. [SICOMP, 2005]Extension to problems w. precedence constraints.
Megow, U. & Vredeveld [MOR, 2006] as well as Chou et al.[2006], Schulz [2008]Stochastic jobs that arrive online.
All for identical machines; use LP lower bound on ΠOPT
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LP Relaxation Identical Machines
Core ingredient: stochastic version of load inequalities[Mohring, Schulz & U., 1999]
∑j∈S
E[Pj ]E[CΠj ] ≥ 1
2m
(∑j∈S
E[Pj ]
)2
+1
2
∑j∈S
E[Pj ]2
−m − 1
2m
∑j∈S
Var[Pj ] ∀ subsets S
Generalizes LPs used earlier [Wolsey, 1985; Queyranne, 1993 &1995; Hall, Schulz, Shmoys & Wein 1997]
But: doesn’t generalize to unrelated machines
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1 Setting the Scene
2 Stochastic Scheduling
3 Unrelated Machines & Time-indexed LP
4 Final Remarks
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Structuring The Input
Theorem
At a cost of (1 + ε), may assume w.l.o.g. input is integer valued.
1
0 time t
Pr[Pj ≥ t]
1 3 4
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Time-Indexed LP Relaxation: Intuition
Instance I and non-anticipatory policy Π, define
xijt := Pr[Π starts job j on machine i at time t ∈ Z≥0]
0 1 2 1110
second blue job, j = 4, has
x1,4,0 = 16/25x2,4,1 = 8/25x1,4,10 = 1/25
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Time-Indexed LP Relaxation
Instance I and non-anticipatory policy Π
xijt := Pr[Π starts job j on machine i at time t ∈ Z≥0]
Properties of xijt (Π non-anticipatory!):
E[Cj ] =∑
i ,t
(t + E[Pij ]
)xijt∑
i ,t xijt = 1 for all jobs j
Pr[i processes j in [s, s + 1]
]=∑s
t=0 xijt Pr[Pij > s − t]
0 1 2 3 4 5 6∑j
s∑t=0
xijt Pr[Pij > s − t] ≤ 1 for each machine i and time s
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Time-Indexed LP Relaxation
min∑
i ,j ,twj
(t + E[Pij ]
)xijt
s.t.∑
i ,txijt = 1 jobs j ,∑
j
∑s
t=0xijt Pr[Pij > s − t] ≤ 1 machines i , times s,
xijt ≥ 0 jobs j , machines i , times t.
Example:
0 1 2 3 4 5 6 7
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Technical Detail: Infinite LP Solution?
Two identical jobs with exponentially distributed processing times:
But: There are feasible LP solutions that are finite, e.g.
Theorem
∃ finite optimal LP solution; LP can be solved efficiently (FPTAS).
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LP-Based Scheduling Policy
Algorithm
1. find an optimal (or approximate) LP solution (xijt);
2. assign each job j independently at random to a machine iwith
Pr[j assigned to i
]=∑
txijt ;
3. apply Smith’s rule on each machine (that’s optimal!);
Theorem (Skutella, Sviridenko & U. 2014)
This algorithm is a ( 3+∆2 )-approximation.
∆ ≥ CV2[Pij ] :=Var[Pij ]
E2[Pij ]for all Pij
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Proof of Performance Ratio
Idea: Analyze more complicated and provably worse algorithm:
1. find an optimal (or approximate) LP solution (xijt);
2. for each job j
a) choose pair (i , t) independently at random with probability xijt ;
b) choose r ∈ Z≥0 indep. at random with probabilityPr[Pij>r ]E[Pij ]
;
c) set the tentative start time of j to s := t + r ;
3. on each machine, sequence jobs by incr. tentative start times;
Example:
0 1 2 3 4 5 6 7
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Proof of Performance Ratio
Key Lemma
Total exp. processing before job j → (i , s) ≤ tent. start time s + 12
Which yields
E[Cj ] ≤∑
i
∑s∈Z≥0
(s + 1
2 + E[Pij ])
Pr[j → (i , s)
]≤ · · · ≤
(3 + ∆
2
)CLPj
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Lower Bounds
Performance bounds are tight . . .
Theorem
Any “fixed assignment” policy can have optimality gap ∆/2.[identical machines].
Theorem
The time-indexed LP can have optimality gap ∆/2.[1 machine].
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1 Setting the Scene
2 Stochastic Scheduling
3 Unrelated Machines & Time-indexed LP
4 Final Remarks
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Final Remarks
To improve, only adaptivity is not enough
[Smith’s rule, even though adaptive, can be as bad as Ω(√
∆)]
open problems are
1. const. approximation (indep. of ∆)?
Im, Moseley & Pruhs [STACS 2015] get
O( log2 n + m log n )− approximation
(identical machines); balancing E[Pj ] vs. Pr[Pj“large”]
2. nontrivial hardness / bounds on approximation ?
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Beating Ω(∆): Im, Moseley, Pruhs 2015
Which jobs first? we want to exclude jobs with. . .
Pr[Pj = “large”] maximal
E[Pj ] maximal
if ALG schedules k jobs A, OPT schedules k jobs A∗, then
Core Lemma
Pr[ALG(A) blocks all machines beyond τ ]≤ Pr[OPT(A∗) blocks all machines beyond τ/α]
α = Θ(m log n)
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Some Reading
Approximation in Stochastic Scheduling:. . .J ACM 46, 1999
When Greediness Fails: Examples from Stochastic SchedulingOR Lett 31, 2003
Stochastic Machine Scheduling with Precedence ConstraintsSIAM Comp 34, 2005
Models and Algorithms for Stochastic Online SchedulingMOR 31, 2006
Unrelated Machine Scheduling with Stochastic Processing TimesMOR 41, 2016
[w/ Megow, Mohring, Schulz, Skutella, Sviridenko, Vredeveld]
Stochastic Scheduling of Heavy-Tailed JobsIm, Moseley & Pruhs, STACS, 2015
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