Generalizing the Kawaguchi-Kyan Bound to Stochastic Parallel Machine Scheduling Sven Jäger Martin Skutella Combinatorial Optimization and Graph Algorithms Technische Universität Berlin 22 nd Combinatorial Optimization Workshop, Aussois
� w j C j Problem P || Given: Weights w j ≥ 0 and processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Process each job nonpreemptively for p j time units on one of the m machines such that the total weighted completion time � n j =1 w j C j is minimized. 2 4 2 5 1 3 1 3 6 5 4 6 0 time C 4 C 1 C 3 C 5 C 2 C 6 S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
The WSPT Rule WSPT rule Whenever a machine becomes idle, start the available job with largest ratio w j / p j on it. The WSPT rule is optimal for a single machine (Smith (1956)) and for unit weights (Conway, Maxwell, Miller (1967)) . S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
The WSPT Rule WSPT rule Whenever a machine becomes idle, start the available job with largest ratio w j / p j on it. The WSPT rule is optimal for a single machine (Smith (1956)) and for unit weights (Conway, Maxwell, Miller (1967)) . Theorem ( Kawaguchi, Kyan (1986) ) √ The WSPT rule is a 1 2 (1 + 2) -approximation, and this bound is tight. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
� w j C j ] Problem P | p j ∼ stoch | E[ Given: Weights w j ≥ 0 and distributions of independent random processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Find a nonpreemptive scheduling policy Π for m identical parallel machines such that the expected weighted sum of completion times is minimized. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
� w j C j ] Problem P | p j ∼ stoch | E[ Given: Weights w j ≥ 0 and distributions of independent random processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Find a nonpreemptive scheduling policy Π for m identical parallel machines such that the expected weighted sum of completion times is minimized. A policy must be nonanticipative, i.e. a decision made at time t may only depend on the information known at time t . S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
The WSEPT Rule WSEPT rule Whenever a machine becomes idle, start the available job with largest ratio w j / E[ p j ] on it. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Known Results − WSEPT has no constant performance guarantee (even for unit weights). (Cheung et al. (2014), Im, Moseley, Pruhs (2015)) S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Known Results − WSEPT has no constant performance guarantee (even for unit weights). (Cheung et al. (2014), Im, Moseley, Pruhs (2015)) + WSEPT is optimal if ◮ there is only one machine (Rothkopf (1966)) , ◮ all jobs have unit weight and processing times are pairwise stochastically comparable (Weber, Varaiya, Walrand (1986)) . S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Known Results − WSEPT has no constant performance guarantee (even for unit weights). (Cheung et al. (2014), Im, Moseley, Pruhs (2015)) + WSEPT is optimal if ◮ there is only one machine (Rothkopf (1966)) , ◮ all jobs have unit weight and processing times are pairwise stochastically comparable (Weber, Varaiya, Walrand (1986)) . + If Var[ p j ] E[ p j ] 2 ≤ ∆ for all j , then WSEPT has performance guarantee 1 + ( m − 1) · (1 + ∆) ≤ 1 + 1 2 · (1 + ∆) . 2 m (Möhring, Schulz, Uetz (1999)) S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Performance Guarantees performance ratio 11 6 2 (1 + ∆) 10 6 1 + 1 9 6 8 6 √ 1 2 (1 + 2) 7 6 1 ∆ 1 1 3 2 0 2 2 S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Performance Guarantees performance ratio 11 6 2 (1 + ∆) 10 6 1 + 1 6 (1 + ∆) 9 1 + 1 6 ) ∆ + 1 ( 1 √ 8 1 ) ∆ + 1 + ( 2 1 6 2 + 1 √ 1 2 (1 + 2) 7 6 1 ∆ 1 1 3 2 0 2 2 S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Performance Guarantees performance ratio 11 6 2 (1 + ∆) 10 6 1 + 1 2 − 1)(1 + ∆) √ 9 this talk: 1 + 1 2 ( 6 8 6 √ 1 2 (1 + 2) 7 6 1 ∆ 1 1 3 2 0 2 2 S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Auxiliary Objective Function Given: Smith ratios ρ j and distributions of independent random processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Find a nonpreemptive scheduling policy for m identical parallel machines such that the expected weighted sum of completion times is minimized, where each job is weighted with its Smith ratio times its actual processing time. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Auxiliary Objective Function Given: Smith ratios ρ j and distributions of independent random processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Find a nonpreemptive scheduling policy for m identical parallel machines such that the expected weighted sum of completion times is minimized, where each job is weighted with its Smith ratio times its actual processing time. ◮ The weight of a job is a random variable w j = ρ j p j . ◮ The Smith ratio ρ j of a job is deterministic. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Auxiliary Objective Function Given: Smith ratios ρ j and distributions of independent random processing times p j ≥ 0 of jobs j = 1 , . . . , n and number m of machines. Task: Find a nonpreemptive scheduling policy for m identical parallel machines such that the expected weighted sum of completion times is minimized, where each job is weighted with its Smith ratio times its actual processing time. ◮ The weight of a job is a random variable w j = ρ j p j . ◮ The Smith ratio ρ j of a job is deterministic. Remark List scheduling the jobs in nonincreasing order of their Smith ratios ρ j is a √ 1 2 (1 + 2)-approximation for the auxiliary objective function. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. Consider auxiliary objective function with weight factors ρ j := w j / E[ p j ]. S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. Consider auxiliary objective function with weight factors ρ j := w j / E[ p j ]. Then, for every policy Π: n � ρ j E[ p j ] E[ C Π Obj(Π) = j ] j =1 original objective function value S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. Consider auxiliary objective function with weight factors ρ j := w j / E[ p j ]. Then, for every policy Π: n n � � ρ j E[ p j ] E[ C Π Obj ′ (Π) ρ j E[ p j C Π Obj(Π) = j ] = j ] j =1 j =1 original objective function value auxiliary objective function value S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. Consider auxiliary objective function with weight factors ρ j := w j / E[ p j ]. Then, for every policy Π: n n � � ρ j E[ p j ] E[ C Π Obj ′ (Π) ρ j E[ p j C Π Obj(Π) = j ] = j ] j =1 j =1 original objective function value auxiliary objective function value E[ p j C Π j ] = E[ p j ( S Π j + p j )] = E[ p j S Π j ] + E[ p 2 j ] S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
Proof of WSEPT’s Performance Guarantee Claim √ The WSEPT rule is a 1 + 1 2 ( 2 − 1) · (1 + ∆)-approximation for P | p j ∼ stoch | E[ � w j C j ]. Consider auxiliary objective function with weight factors ρ j := w j / E[ p j ]. Then, for every policy Π: n n � � ρ j E[ p j ] E[ C Π Obj ′ (Π) ρ j E[ p j C Π Obj(Π) = j ] = j ] j =1 j =1 original objective function value auxiliary objective function value E[ p j C Π j ] = E[ p j ( S Π j + p j )] = E[ p j S Π j ] + E[ p 2 j ] j ] + E[ p j ] 2 + Var[ p j ] = E[ p j ] E[ C Π = E[ p j ] E[ S Π j ] + Var[ p j ] . nonanticipativity S. Jäger, M. Skutella (TU Berlin) Generalizing the KK Bound to Stoch. Scheduling C.O.W. 2018
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