Greedy Algorithms for Stochastic Scheduling on Unrelated Machines - - PowerPoint PPT Presentation

Greedy Algorithms for Stochastic Scheduling on Unrelated Machines

Marc Uetz m.uetz@utwente.nl

joint work with Varun Gupta Ben Moseley Qiaomin Xie

This Talk

Analyses of algorithms stochastic scheduling:

• 1. either offline
• 2. or restricted to identical machines
• 3. or required (sophisticated) LP-relaxations

this talk first performance bounds for simple greedy algorithm:

• online
• unrelated machines
• stochastic jobs

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1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks

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Single Machine Scheduling

Given: n jobs j, weights wj > 0, nonpreemptive processing times pj ∈ Z>0 Task: sequence jobs on 1 machine; one job at a time; minimize

j wj Cj where Cj = j’s completion time;

time Cred Cgreen Corange Cblue

Theorem (Smith 1956)

Smith’s rule, sequencing jobs in order wj/pj ց 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

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

time

Theorem

Problem is APX-hard [Hoogeveen et al., 2002] Exists ( 3

2 − c)-approximation [Bansal, Srinivasan, Svensson, 2016]

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Uncertainty in Scheduling

Uncertainty of job sizes pj (or pij): non-clairvoyant online models, w. preemption allowed [many, many references] no preemption: non-clairvoyant Ω( n )-competitive Here: no preemption, but with probabilistic info on pj

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1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks

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Stochastic Scheduling

job size = (independent) random variables Pj (or Pij); all known 1 time t Pr[Pj ≥ t]

Solution: Non-anticipatory scheduling policy Π

Decisions based on information up to now and a priori knowledge about Pj (or Pij); no further information about the future.

time now

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Optimality

Π(I) := cost of policy Π on instance I, is a random variable

Definition (Optimal Policy)

Call ΠOPT optimal if it achieves inf{ E[Π(I)] | Π non-anticipatory policy }

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Example

n = 4 jobs, weights wj = 1 time 1 10 blue jobs: Pj = 1 green jobs: Pj =

• probability 4/5

10 probability 1/5 (note E[Pj] = 2) Schedule on m = 2 identical machines.

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Stochastic World

Unique optimal policy: Start green + blue

Then continue:

• green → blue

if first green job short blue → green if first green job long [with Π(I) = E[

j Cj] = 6.76].

1 2 11 12 10 Complicated tradeoff between large E[Pj] or large Pr(Pj = “∞”) (i.e., heavy tail) Even deliberate idleness may be necessary [U. 2003]

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Approximation Algorithms

Optimal policies

intuitively complex, exponential size decision tree; definitely NP(APX)-hard, . . .

• nly 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)] Adversary is non-anticipatory, too!

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Approximation Algorithms

M¨

• hring, Schulz & U. [JACM, 1999]

First LP-based approximation algorithms e.g.: Smith’s rule has performance guarantee ( 3+∆

2 ).

Skutella & U. [SICOMP, 2005], Megow, U. & Vredeveld [Math. OR, 2006] Chou et al. [OR 2006], Schulz [2008]

Problems w. precedence constraints, jobs that arrive online.

Moseley, Im, Pruhs [STACS, 2015]

O( log2 n + m log n )-approximation All results ↑ for identical machines

Skutella, Sviridenko, U. [Math. OR, 2016]

Guarantee ( 3+∆

2 ) for unrelated machines, time-indexed LP

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1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks

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Main Result

Model 1: Stochastic jobs appear one after another (t = 0), must be assigned to machine upon arrival

Theorem

Greedy online algorithm has a performance guarantee (8 + 4∆). ∆ = upper bound on the (squared) coeff. of variation CV[Pij] := Var[Pij] / E2[Pij] ≤ ∆ for all Pij Model 2: stochastic jobs arrive over time (at release times rj), performance guarantee is (144 + 72∆)

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Algorithm

Algorithm Greedy

per machine, jobs sequenced as in Smith’s rule (wj/E[Pij]) when job j arrives (order 1, 2, . . . ), compute for each machine i ∈ M expected instantaneous increase in objective, i.e., wjE[Pij] + wj

• k<j,k→i,k∈H(j,i)

E[Pik] + E[pij]

• k<j,k→i,k∈L(j,i)

wk . assign job j to any machine minimizing this quantity i j H(j, i) L(j, i)

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Rough Sketch Analysis

• 1. set up LP Relaxation (for stochastic problem)
• 2. simplify LP relaxation

– losing O( ∆ )

• 3. analysis of Greedy using dual solution – losing O( 1 )

called “dual fitting” by [Anand et al., SODA 2012]

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1 - Time-Indexed LP Relaxation

Instance I and non-anticipatory policy Π, yijt := Pr[Π has job j in process on machine i at [t, t + 1)] 1 2 11 12 10 second, blue job, j = 4, has y1,4,0 = 16/25 y2,4,1 = 9/25

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1 - Time-Indexed LP Relaxation

Instance I and non-anticipatory policy Π, yijt := Pr[Π has job j in process on machine i at [t, t + 1)] Properties of yijt :

• i∈M
• t≥0

yijt E[Pij] = 1

for all j ∈ J [Π non-anticipatory!]

• j∈J yijt ≤ 1

for all i ∈ M, t ≥ 0 with some calculus, one shows that E[Cj] =

i∈M

• t≥0

yijt

E[Pij]

• t + 1

2

• + 1−CV[Pij]2

2

yijt

• for all j ∈ J

E[Cj] ≥

i∈M

• t≥0 yijt [for analysis]

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1 - LP Relaxation (for stochastic problem)

zS := min

• j∈J

wj C S

j

s.t. C S

j =

• i∈M
• t≥0

yijt E[Pij]

• t + 1

2

• + 1 − CV[Pij]21 − CV[Pij]2

2 yijt

• i∈M
• t≥0

yijt E[Pij] = 1 ∀ jobs j,

• j∈J

yijt ≤ 1 ∀ machines i, times t,

• i∈M
• t≥0

yijt ≤ Cj ∀ jobs j, yijt ≥ 0 ∀ jobs j, machines i, times t.

Would like to work with (LP) dual, but. . .

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2 - Simplified LP Relaxation

zP := min

• j∈J

wj C P

j

s.t. C P

j =

• i∈M
• t≥0

yijt E[Pij]

• t + 1

2

• + 1

2 yijt

• i∈M
• t≥0

yijt E[Pij] = 1 jobs j,

• j∈J

yijt ≤ 1 machines i, times t, yijt ≥ 0 jobs j, machines i, times t.

Lemma

zP ≤

• 1 + ∆

2

• zS

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2 - LP Dual

the dual has variables (α, β); max zD =

• j∈J

αj −

• i∈M
• t≥0

βis s.t. αj ≤ E[Pij]βi,t + wj

• t + 1

2 + E[Pij] 2

• for all i, j, t

βit ≥ 0 for all i, t Goal: dual solution that relates to Greedy

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3 - Greedy & Dual LP

Interpreting the Dual

αj := expected increase (of Greedy) of objective upon arrival of job j βi,t := total expected weight of jobs assigned to machine i, and still unfinished at time t (by Greedy) Can show

Lemma

Solution (α/2, β/2) is dual feasible. Yet of little help, as zD =

j∈J αj − i∈M

• t≥0 βi,t, and
• j∈J αj =
• bjective value of Greedy
• i∈M
• t≥0 βit = objective value of Greedy, too

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3 - Speed Augmentation

Alternative dual LP solution

αj ˜ αj := expected increase (of Greedy) of objective upon arrival of job j βi,t ˜ βi,t := total expected weight of jobs assigned to machine i, and still unfinished at time t (by Greedy) For the same instance, but assuming all machines run at speed 2. Now can show

Lemma

Solution (˜ α/2, ˜ β/4) is feasible (for original LP) Proof: as ˜ α = 1

2α and ˜

βi,t = βi,2t . . . by dual constraint. . .

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3 - Putting Things Together

Now we have zD(˜ α/2, ˜ β/4) = 1 2

• j∈J

˜ αj − 1 4

• i∈M
• t≥0

˜ βi,t = 1 4Greedy − 1 8Greedy So Greedy = 8zD(˜ α/2, ˜ β/4) ≤ 8zD = 8zP ≤ 8(1 + ∆ 2 )zS ≤ (8 + 4∆)OPT

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1 Setting the Scene 2 Stochastic Scheduling 3 Stochastic Online Scheduling, Unrelated Machines 4 Final Remarks

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Final Remarks

O( ∆ ) is tight (for greedy), there is a ∆/2 lower bound jobs released over time (rj) requires more work, but doable

• pen problems
• 1. is const. approximation (indep. of ∆) possible?

Identical machines: Megow & Vredeveld [MOR 2014]: 2-approximation (w. preemption allowed) Im, Moseley & Pruhs [STACS 2015]: O( log2 n + m log n )-approximation (w/o preemption)

• 2. not much known on hardness or lower bounds for

approximation

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thanks for your attention!

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