Online Non-preemptive Scheduling in a Resource Augmentation Model - PowerPoint PPT Presentation

Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Giorgio Lucarelli 1 Nguyen Kim Thang 2 Abhinav Srivastav 1 Denis Trystram 1 1 LIG, University of Grenoble-Alpes 2 IBISC, University of Evry Val d’Essonne New Challenges in Scheduling Theory, 2016 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 1 / 23

Problem definition Instance: a set of m unrelated machines M , a set of n jobs J , and for each job j ∈ J : - a machine-dependent processing time p ij - a release date r j - a weight w j Goal: a non-preemptive schedule that minimizes total weighted flow time: � w j ( C j − r j ) j ∈J where C j is the completion time of job j ∈ J G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 2 / 23

Problem definition Instance: a set of m unrelated machines M , a set of n jobs J , and for each job j ∈ J : - a machine-dependent processing time p ij - a release date r j - a weight w j Goal: a non-preemptive schedule that minimizes total weighted flow time: � w j ( C j − r j ) j ∈J where C j is the completion time of job j ∈ J Setting jobs arrive online job characteristics ( p ij , w j ) become known after the release of j G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 2 / 23

Previous work for non-preemptive scheduling Offline Lower bound: Ω( n 1 / 2 − ǫ ) even on a single machine for total (unweighted) flow time [Kellerer et al. 1999] O ( � n m log n m ) -approximation algorithm for identical machines to minimize total (unweighted) flow time [Leonardi and Raz 2007] Online Lower bound: Ω( n ) even on a single machine for total (unweighted) flow time [Chekuri et al. 2001] Θ( p max p min + 1) -competitive algorithm for a single machine [Tao and Liu 2013] G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 3 / 23

Resource augmentation The algorithm is allowed to use more resources than the optimal use highest speed [Phillips et al. 1997, Kalyanasundaram and Pruhs 2000] use more machines [Phillips et al. 1997] G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 4 / 23

Resource augmentation The algorithm is allowed to use more resources than the optimal use highest speed [Phillips et al. 1997, Kalyanasundaram and Pruhs 2000] use more machines [Phillips et al. 1997] reject jobs [Choudhury et al. 2015] G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 4 / 23

Resource augmentation The algorithm is allowed to use more resources than the optimal use highest speed [Phillips et al. 1997, Kalyanasundaram and Pruhs 2000] use more machines [Phillips et al. 1997] reject jobs [Choudhury et al. 2015] Refined competitive ratio: algorithm’s solution using resource augmentation offline optimal solution (without resource augmentation) G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 4 / 23

Previous work (cont’d) Offline 12-speed 4-approximation algorithm for a single machine [Bansal et al. 2007] (1 + ǫ ) -speed (1 + ǫ ) -approximation quasi-polynomial time algorithm for identical machines [Im et al. 2015] Online O (log p max p min ) -machines O (1) -competitive for identical machines [Phillips et al. 1997] O (log n ) -machine O (1) -speed 1-competitive for total (unweighted) flow time on identical machines [Phillips et al. 1997] √ n } ) -competitive algorithm for total � p max ℓ -machines O (min { ℓ p min , ℓ (unweighted) flow time on a single machine [Epstein and van Stee 2006] optimal up to a constant factor for constant ℓ G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 5 / 23

Our contribution � p max Lower bound: for any speed augmentation s ≤ p min , every deterministic 10 � p max algorithm has competitive ratio at least Ω( 10 p min ) even for a single machine G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 6 / 23

Our contribution � p max Lower bound: for any speed augmentation s ≤ p min , every deterministic 10 � p max algorithm has competitive ratio at least Ω( 10 p min ) even for a single machine Resource augmentation algorithms (1 + ǫ s ) -speed ǫ r -rejection 2(1+ ǫ r )(1+ ǫ s ) -competitive algorithm ǫ r ǫ s extension for ℓ k -norms G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 6 / 23

Linear programming formulation Definitions δ ij = w j p ij : density of the job j on machine i R : set of rejected jobs � 1, if job j is executed on machine i at time t variable x ij ( t ) = 0, otherwise G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 7 / 23

Linear programming formulation Definitions δ ij = w j p ij : density of the job j on machine i R : set of rejected jobs � 1, if job j is executed on machine i at time t variable x ij ( t ) = 0, otherwise Lower bounds to our objective fractional flow time of job j : � ∞ δ ij ( t − r j ) x ij ( t ) dt r j weighted processing time of job j � ∞ � ∞ w j p j = w j x ij ( t ) dt = δ ij p ij x ij ( t ) dt r j r j G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 7 / 23

Linear programming relaxation Primal � ∞ � � min δ ij ( t − r j + p ij ) x ij ( t ) dt r j i ∈M j ∈J � ∞ x ij ( t ) � dt ≥ 1 ∀ j ∈ J p ij r j i ∈M � x ij ( t ) ≤ 1 ∀ i ∈ M , t ≥ 0 j ∈J x ij ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 8 / 23

Linear programming relaxation Primal � ∞ � � min δ ij ( t − r j + p ij ) x ij ( t ) dt r j i ∈M j ∈J � ∞ x ij ( t ) � dt ≥ 1 ∀ j ∈ J p ij r j i ∈M � x ij ( t ) ≤ 1 ∀ i ∈ M , t ≥ 0 j ∈J x ij ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 Dual � ∞ � � max λ j − γ i ( t ) dt 0 j ∈J i ∈M λ j p ij − γ i ( t ) ≤ δ ij ( t − r j + p ij ) ∀ i ∈ M , j ∈ J , t ≥ r j λ j , γ i ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 8 / 23

Speed interpretation Primal � ∞ � � min δ ij ( t − r j + p ij ) x ij ( t ) dt r j i ∈M j ∈J � ∞ x ij ( t ) � dt ≥ 1 ∀ j ∈ J p ij r j i ∈M 1 � x ij ( t ) ≤ ∀ i ∈ M , t ≥ 0 1 + ǫ s j ∈J x ij ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 Dual � ∞ 1 � � max λ j − γ i ( t ) dt 1 + ǫ s 0 j ∈J i ∈M λ j p ij − γ i ( t ) ≤ δ ij ( t − r j + p ij ) ∀ i ∈ M , j ∈ J , t ≥ r j λ j , γ i ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 9 / 23

Rejection interpretation Primal � ∞ � � min δ ij ( t − r j + p ij ) x ij ( t ) dt r j i ∈M j ∈J \R � ∞ x ij ( t ) � dt ≥ 1 ∀ j ∈ J \ R p ij r j i ∈M � x ij ( t ) ≤ 1 ∀ i ∈ M , t ≥ 0 j ∈J \R x ij ( t ) ≥ 0 ∀ i ∈ M , j ∈ J \ R , t ≥ 0 Dual � ∞ � � max λ j − γ i ( t ) dt 0 i ∈M j ∈J \R λ j p ij − γ i ( t ) ≤ δ ij ( t − r j + p ij ) ∀ i ∈ M , j ∈ J \ R , t ≥ r j λ j , γ i ( t ) ≥ 0 ∀ i ∈ M , j ∈ J \ R , t ≥ 0 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 10 / 23

Competitive ratio Primal � ∞ � � min δ ij ( t − r j + p ij ) x ij ( t ) dt r j i ∈M j ∈J \R � ∞ x ij ( t ) � dt ≥ 1 ∀ j ∈ J \ R p ij r j i ∈M � x ij ( t ) ≤ 1 ∀ i ∈ M , t ≥ 0 j ∈J \R x ij ( t ) ≥ 0 ∀ i ∈ M , j ∈ J \ R , t ≥ 0 Dual � ∞ 1 � � max λ j − γ i ( t ) dt 1 + ǫ s 0 j ∈J i ∈M λ j p ij − γ i ( t ) ≤ δ ij ( t − r j + p ij ) ∀ i ∈ M , j ∈ J , t ≥ r j λ j , γ i ( t ) ≥ 0 ∀ i ∈ M , j ∈ J , t ≥ 0 G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 11 / 23

Competitive ratio � ∞ � � δ ij ( t − r j + p ij ) x ij ( t ) dt r j Primal ( speed = 1 , J \ R ) i ∈M j ∈J \R = � ∞ 1 1 Dual ( speed = 1+ ǫ s , J ) � � λ j − γ i ( t ) dt 1 + ǫ s 0 j ∈J i ∈M G. Lucarelli Online Non-preemptive Scheduling in a Resource Augmentation Model based on Duality Aussois 2016 12 / 23

Intuition of rejection time 0 P

Intuition of rejection time 0 1 P

Intuition of rejection time 0 1 P P + 1

Intuition of rejection time 0 1 2 P P + 1

Intuition of rejection time 0 1 2 3 P P + 1

Intuition of rejection . . . time 0 1 2 3 P P + 1 2 P P small jobs each small job has flow time P

Intuition of rejection . . . time 0 1 2 3 P P + 1 2 P time 0 1 2 3 P P + 1 2 P + 1 P small jobs each small job has flow time P ... while in the optimal it has flow time 1