Greed Is Good (For Scheduling Under Uncertainty) · Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz [email protected] (updated) IPCO 2017 paper, with: Varun Gupta, Ben

Greed Is Good(For Scheduling Under Uncertainty)

Marc [email protected]

(updated) IPCO 2017 paper, with:

Varun Gupta, Ben Moseley, Qiaomin Xie

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 enoughfirst 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 ]

Marc Uetz - Greed. . . is good 2

“Greed, . . . , greed is good!”


Unrelated Machine Scheduling – R | |∑

wjCj

Given: m machines, n non-preemptive jobs with weights wj andmachine-dependent processing times pij :

Cj := completion time job j in schedule; minimize∑

j wj Cj

0 timeCblue

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]


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.]


1 Stochastic Scheduling

2 Greedy Algorithm for Unrelated Machines

3 Final Remarks


Stochastic Processing Times

job j appears: size = (independent) random variables Pij ; known

1

0time t

Pr[Pij ≥ t]

Solution: Non-anticipatory scheduling policy Π

Decisions may only use information up to now.

0 timenow

Objective: Min. expected performance E[∑

wjCj ] =∑

wjE[Cj ]


Example with Four Jobs

n = 4 jobs, all weights wj = 1

time0 1 10

blue jobs: Pj = 1

green jobs: Pj =

0 probability 4/5

10 probability 1/5(E[Pj ] = 2)

Optimal policy for m = 2 identical machines?


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]

0 1 2 11 1210

Complicated tradeoff between

small E[Pj ] or large Pr(Pj = 0) (but, heavy tail)


Approximation for Stochastic Scheduling

Optimal policies hopeless, even offline, . . .

Definition (Approximation)

Policy Π has performance guarantee α ≥ 1, if for allinstances P

E[∑

wjCΠj ] ≤ α E[

∑wjC

OPTj ]

Adversary OPT knows set of jobs, but subject to uncertainprocessing times Pij , too

Classical competitive analysis is special case


Coefficient of Variation

Performance guarantees depend on “variability” of Pij

Define

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


Approximation Algorithms Stochastic Scheduling

For identical machines:

Mohring, Schulz & U. [J.ACM, 1999]

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

2 )

improved to 1 + 12 (√

2− 1)(1 + ∆) [Jager & 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


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


Greedy Algorithm I

sequencing: available jobs w. max. wj/E[Pij ] go first

assignment: use “proxy” for E[increase] of objective: job jappears ar rj , consider all jobs that arrived earlier, theirmachine assignments fixed, and assuming pij = E[Pij ] andrk = 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

0

i


Greedy Algorithm II

sequencing: available jobs w. max. wj/E[Pij ] go first

availability: job j declared available on machine i at slightlyinflated release time rij ≥ rj

rij = maxE[Pij ], sj

where sj = start time of job j in nominal Greedy schedulewhere we assume pik = E[Pik ] for all jobs on i[note: this can be computed online]

rj rij

j

0

i


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]


1 - Time-Indexed LP

yijt := Pr[machine i has job j in process at [t, t + 1)]

0 1 2 11 1210

second, green job (say j = 3), has at time t = 2

y1,3,2 = 4/25y2,3,2 = 1/25


1 - LP Relaxation (Not a Formulation!)

zS := min∑j∈J

wj CSj

s.t. CSj =

∑i∈M

∑t≥0

(yijt

E[Pij ]

(t + 1

2

)+

1− CV[Pij ]21− CV[Pij]

2

2yijt

)∑i∈M

∑t≥rj

yijtE[Pij ]

= 1 ∀ jobs j ,

∑j∈J

yijt ≤ 1 ∀ machines i , times t,

∑i∈M

∑t≥rj

yijt ≤ CSj ∀ jobs j ,

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

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


2 - Simplified LP Relaxation

zP := min∑j∈J

wj CPj

s.t. CPj =

∑i∈M

∑t≥0

(yijt

E[Pij ]

(t + 1

2

)+

1

2yijt

)∑i∈M

∑t≥0

yijtE[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


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 feasibledual solution (α, β) with

zD(α, β) =1

6Greedydet

[idea: 3x speed augmentation]Marc Uetz - Greed. . . is good 20

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

sdetj

sdetj = starting time job j in Greedydet

Proof:

Sj ≤ sj +∑

predecessors k

(Pk − E[Pk ])+ . . .


Final Remarks

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

open problems

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

2. is stochastic problem harder to approximate ?

thanks for your attention!


Documents

Greed Is Good (For Scheduling Under Uncertainty) · Greed Is Good (For Scheduling Under Uncertainty) Marc Uetz [email protected] (updated) IPCO 2017 paper, with: Varun Gupta, Ben