combinatorial algorithms for minimizing the weighted sum ......singleiteration(jobsetj) increase j...

Combinatorial Algorithms for Minimizing the WeightedSum of Completion Times on a Single Machine

James Davis 1 Rajiv Gandhi2 Vijay Khothari

1Department of Operations ResearchCornell University

[email protected]

2Department of Computer ScienceRutgers University - Camden

[email protected]

November 8, 2010

James Davis (Cornell University) Machine Scheduling 1 / 22

Single Machine Scheduling (Off-line Problem)

Input:J, set of jobspj , processing timesrj ≥ 0, release dateswj , weights

Goal:Schedule jobs on a single machineNon-preemptive scheduleNo job can be scheduled before it’s release dateMinimize

∑j wjCj , weighted sum of completion times



Cost = 1 · 1+ 6 · 8+ 9 · 4+ 13 · 3


Single Machine Scheduling (Online Problem)

Input:Jobs, Jpj , rj , wj ∀j ∈ J

Goal:Schedule jobs on a single machineNon-preemptive scheduleMinimize

∑j wjCj , weighted sum of completion times

Aware of job j at time rjSchedule fixed at time t without knowledge of jobs s.t. rj > t


LP Formulation

Notation (S ⊆ J):Sum of processing times

p(S) =∑j∈S

pj

Sum of squared processing times,

p2(S) =∑j∈S

p2j


Simple LP Formulation

Cj denotes the completion time of job j

min∑j∈J

wjCj

subject to Cj ≥ rj + pj, ∀j ∈ J∑j∈S

pjCj ≥p(S)2 + p2(S)

2, ∀S ⊆ J

Cj ≥ 0, ∀j ∈ J


Previous Work

Single Machine Scheduling:NP-Hard (Lenstra et al.)Off-line PTAS known (Afrati et al.)Off-line 1.6853-Appx. Alg. via LP rounding (Goemans et al.)Online 1.6853-Appx. Alg. via LP rounding (Goemans et al.)

Using Simple LP (provide upper bound on int. gap):Off-line 3-Appx. Alg. via LP rounding (Hall et al.)


Our Contribution

Use Simple LP (provide upper bound on int. gap):Off-line (1+

√2)(≈ 2.42)-Approximation Algorithm

Online 3-Approximation Algorithm


Main Criteria

Recall Notation:p(S) =

∑j∈S pj

New Notation:j∗ ∈ J has highest release date (rj value)j′ ∈ J has lowest wjpj value

Main Criteriarj∗ > p(J)


List Algorithm

Off-line List AlgorithmJ ′ ← Jwhile J ′ 6= ∅ do

j∗ ← job with largest rj valueif rj∗ > p(J ′) thenRemove j∗ from J ′

else if rj∗ ≤ p(J ′) thenj′ ← j ∈ J ′ with lowest wjpj valueRemove j′ from J ′

end ifend whileSchedule jobs in the reverse order that they were removed from J ′


Intuitive Analysis

If rj∗ > p(J ′) then Cj∗ is approximately dominated by rj∗

rj∗ = 11 > 10 = p(J ′)


Intuitive Analysis

If rj∗ ≤ p(J ′) then Cj∗ is approximately dominated by p(J ′)

rj∗ = 2 ≤ 8 = p(J ′)


Intuitive Analysis

Smith’s RuleSchedule the most useless job last (lowest wjpj )

j′ is job 1 (w1p1 = 1)


Intuitive Analysis

Final Schedule


Simple LP Formulation

Cj denotes the completion time of job j

min∑j∈J

wjCj

subject to Cj ≥ rj + pj, ∀j ∈ J∑j∈S

pjCj ≥p(S)2 + p2(S)

2, ∀S ⊆ J

Cj ≥ 0, ∀j ∈ J


Simple Dual

Introduce βS ∀S ⊆ J and αj ∀j ∈ J

max∑j∈J

αj(rj + pj)+∑S⊆J

βS

(p(S)2 + p2(S)2

)subject to αj + pj

∑S:j∈S

βS ≤ wj, ∀j ∈ J

αj ≥ 0, ∀j ∈ JβS ≥ 0, ∀S ⊆ J


Primal-Dual AlgorithmSimplified Setting:

Single Iteration (job set J)Increase αj∗ or βJ

rj∗ > p(J ′)Increase αj∗αj∗ = wj∗

rj∗ ≤ p(J ′)Increase βJβJ =

wj′pj′

Dual Constraintαj + pj

∑S:j∈S βS ≤ wj


Primal-Dual Algorithm

Off-line Primal-Dual AlgorithmJ ′ ← Jwhile J ′ 6= ∅ do

j∗ ← job with largest rj valueif rj∗ > p(J ′) thenαj∗ ← wj∗ − pj∗

∑S:j∗∈S

βS

Remove j∗ from J ′else if rj∗ ≤ p(J ′) then

j′ ← j ∈ J ′ with lowest wjpj valueβJ′ ←

wj′pj′−∑

S:j∈S βS

Remove j′ from J ′end if

end whileSchedule jobs in the reverse order that they were removed from J ′


Primal-Dual Algorithm

TheoremUsing the off-line primal dual algorithm:

Cost of Schedule ≤ (1+√2) · Dual Feasible Solution ≤ (1+

√2) · OPT


Thank You!


combinatorial algorithms for minimizing the weighted sum ......singleiteration(jobsetj) increase j...

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