Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz - - PowerPoint PPT Presentation

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Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz - - PowerPoint PPT Presentation

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Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz m.uetz@utwente.nl (updated) IPCO 2017 paper, with: Varun Gupta, Ben Moseley, Qiaomin Xie 1-Slide Overview Talk is about (basic, notorious) stochastic scheduling problem [defined

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Greed Is Good (For Scheduling Under Uncertainty)

Marc Uetz m.uetz@utwente.nl

(updated) IPCO 2017 paper, with: Varun Gupta, Ben Moseley, Qiaomin Xie

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1-Slide Overview

Talk is about (basic, notorious) stochastic scheduling problem [defined later] Almost all previous results (scheduling in general): good algorithms based on solving (complicated LP) relaxations Alternative title: To hell with (LP-)relaxations! if greedy (online) algorithms are good enough first results (for this model) where jobs appear online Greedy algorithm has performance guarantee (144 + 72∆)(72 + 36∆)(12 + 6∆)(6 + 3∆)h(∆)

[∆ = (squared) coeff. variation, 1 ≤ h( · ) ≤ 2 with h(0) = 1 ]

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“Greed, . . . , greed is good!”

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Unrelated Machine Scheduling – R | | wjCj

Given: m machines, n non-preemptive jobs with weights wj and machine-dependent processing times pij: Cj := completion time job j in schedule; minimize

j wj Cj

time Cblue

Theorem

Offline: problem is APX-hard [Hoogeveen et al., 2002] Offline: LP-based 1.4994-approximation [Li, 2018] Online: lower bound 1.309 [Vestjens 1997]

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Uncertainty in Scheduling

Job j appears at release time rj Non-clairvoyant online models (job lengths unknown till Cj)

  • hopeless, since jobs cannot be preempted

[Ω(n) even in simplest settings] Clairvoyant online models (job lengths known upon arrival rj)

  • deterministic algo.: 8-competitive

[Hall et al. MOR 1997]

  • randomized algo.: 5.78-competitive

[Chakrabarti et al. ICALP 1996]

Stochastic Online Model (= Data Science)

Non-clairvoyant, but probabilistic info on pij [our result ⇒ Greedy is deterministic 6-competitive algo.]

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1 Stochastic Scheduling 2 Greedy Algorithm for Unrelated Machines 3 Final Remarks

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Stochastic Processing Times

job j appears: size = (independent) random variables Pij; known 1 time t Pr[Pij ≥ t]

Solution: Non-anticipatory scheduling policy Π

Decisions may only use information up to now.

time now

Objective: Min. expected performance E[ wjCj] = wjE[Cj]

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Example with Four Jobs

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

  • probability 4/5

10 probability 1/5 (E[Pj] = 2) Optimal policy for m = 2 identical machines?

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Policies are Complex (and Dynamic)

Unique optimal policy: Start green + blue

Then continue:

  • first green then blue

if first green job = 0 first blue then green if first green job = 10 [with E[

j wjCj] = 6.76]

1 2 11 12 10 Complicated tradeoff between small E[Pj] or large Pr(Pj = 0) (but, heavy tail)

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Approximation for Stochastic Scheduling

Optimal policies hopeless, even offline, . . .

Definition (Approximation)

Policy Π has performance guarantee α ≥ 1, if for all instances P E[

  • wjC Π

j ] ≤ α E[

  • wjC OPT

j

] Adversary OPT knows set of jobs, but subject to uncertain processing times Pij, too Classical competitive analysis is special case

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Coefficient of Variation

Performance guarantees depend on “variability” of Pij

Define

∆ := maxi,j CV[Pij]2 = Var[Pij] / E2[Pij]

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Approximation Algorithms Stochastic Scheduling

For identical machines:

M¨

  • hring, Schulz & U. [J.ACM, 1999]

first LP-based approximation algorithm e.g.: Smith’s rule (wj/E[Pj] ց) has guarantee ( 3+∆

2 )

improved to 1 + 1

2(

√ 2 − 1)(1 + ∆) [J¨ ager & Skutella 2018]

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

Problems w. precedence constraints, release times, or online

Im, Moseley, Pruhs [STACS, 2015]

remarkable O( log2 n + m log n )-approximation For unrelated machines:

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

( 3+∆

2 ) for offline, using time-indexed LP relaxation

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Greedy for Unrelated Machines, Stochastic Jobs

Theorem

If all release times rj = 0 (“online-list”), Greedy has performance guarantee 4 + 2∆, analysis tight for ∆ = 0 [Correa & Queyranne, 2012].

Theorem

In general (release times rj > 0, “online-time”), Greedy has performance guarantee (6 + 3∆)h(∆) . 1 ≤ h( · ) ≤ 2 with h(0) = 1, h(1) = 3/2

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Greedy Algorithm I

sequencing: available jobs w. max. wj/E[Pij] go first assignment: use “proxy” for E[increase] of objective: job j appears ar rj, consider all jobs that arrived earlier, their machine assignments fixed, and assuming pij = E[Pij] and rk = 0 for all jobs, compute expected increase of

j wjE[Cj]

if j was inserted into sequence in order of wj/E[Pij] → assign job j to any machine minimizing this increase j i

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Greedy Algorithm II

sequencing: available jobs w. max. wj/E[Pij] go first availability: job j declared available on machine i at slightly inflated release time rij ≥ rj rij = max{E[Pij], sj} where sj = start time of job j in nominal Greedy schedule where we assume pik = E[Pik] for all jobs on i [note: this can be computed online] rj rij j i

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Sketch Analysis

  • 1. time-indexed LP Relaxation (for stochastic problem) - sorry
  • 2. “simplify” that LP relaxation

– losing O( ∆ )

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

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

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1 - Time-Indexed LP

yijt := Pr[machine i has job j in process at [t, t + 1)] 1 2 11 12 10 second, green job (say j = 3), has at time t = 2 y1,3,2 = 4/25 y2,3,2 = 1/25

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1 - LP Relaxation (Not a Formulation!)

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≥rj

yijt E[Pij] = 1 ∀ jobs j,

  • j∈J

yijt ≤ 1 ∀ machines i, times t,

  • i∈M
  • t≥rj

yijt ≤ C S

j

∀ 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

βit s.t. αj ≤ E[Pij]βit + wj

  • t + E[Pij]

2 + 1 2

  • for all i, j, t

βit ≥ 0 for all i, t

Lemma

Considering Greedydet (for pij = E[pij]); can construct feasible dual solution (˜ α, ˜ β) with zD(˜ α, ˜ β) = 1 6Greedydet [idea: 3x speed augmentation]

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3 - Final Steps

By duality Greedydet = 6zD(˜ α, ˜ β) ≤ 6zD=6zP ≤ 6(1 + ∆ 2 )zS ≤ (6 + 3∆)OPT Finally, can show for (true) expected starting time job j

Lemma

E[Sj] ≤ h(∆)

≤2

sdet

j

sdet

j

= starting time job j in Greedydet Proof: Sj ≤ sj +

  • predecessors k

(Pk − E[Pk])+ . . .

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Final Remarks

O( ∆ ) is tight (for greedy), there is a ∆/2 lower bound

  • pen problems
  • 1. is const. approximation (indep. of ∆) possible?
  • 2. is stochastic problem harder to approximate ?

thanks for your attention!

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