# Scheduling Parallel Machine Scheduling. Tim Nieberg

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1 Scheduling Parallel Machine Scheduling Tim Nieberg

2 Problem P C max : m machines n jobs with processing times p 1,..., p n

3 Problem P C max : m machines n jobs with processing times p 1,..., p { n 1 if job j is processed on machine i variable x ij = 0 else ILP formulation: min s.t. C max n j=1 x ijp j C max i = 1,..., m m i=1 x ij = 1 j = 1,..., n x ij {0, 1} i = 1,..., m; j = 1,..., n

4 Problem P C max : in lecture 2: P2 C max is NP-hard P C max is even NP-hard in the strong sense (reduction from 3-PARTITION); i.e. also pseudopolynomial algorithms are unlikely question: What happens if x ij {0, 1} in the ILP is relaxed?

5 Problem P C max : in lecture 2: P2 C max is NP-hard P C max is even NP-hard in the strong sense (reduction from 3-PARTITION); i.e. also pseudopolynomial algorithms are unlikely question: What happens if x ij {0, 1} in the ILP is relaxed? answer: objective value of LP gets n j=1 p j/m question: is this the optimal value of P pmtn C max?

6 Problem P C max : in lecture 2: P2 C max is NP-hard P C max is even NP-hard in the strong sense (reduction from 3-PARTITION); i.e. also pseudopolynomial algorithms are unlikely question: What happens if x ij {0, 1} in the ILP is relaxed? answer: objective value of LP gets n j=1 p j/m question: is this the optimal value of P pmtn C max? answer: No! Example: m = 2, n = 2, p = (1, 2)

7 Problem P C max : in lecture 2: P2 C max is NP-hard P C max is even NP-hard in the strong sense (reduction from 3-PARTITION); i.e. also pseudopolynomial algorithms are unlikely question: What happens if x ij {0, 1} in the ILP is relaxed? answer: objective value of LP gets n j=1 p j/m question: is this the optimal value of P pmtn C max? answer: No! Example: m = 2, n = 2, p = (1, 2) add C max p j for j = 1,..., m to ensure that each job has enough time

8 LP for problem P pmtn C max : min s.t. C max n x ij p j C max i = 1,..., m j=1 p j C max j = 1,..., n m x ij = 1 j = 1,..., n i=1 x ij 0 i = 1,..., m; j = 1,..., n

9 LP for problem P pmtn C max : min s.t. C max n x ij p j C max i = 1,..., m j=1 p j C max j = 1,..., n m x ij = 1 j = 1,..., n i=1 x ij 0 i = 1,..., m; j = 1,..., n Optimal value of LP is max{max n j=1 p j, n j=1 p j/m} LP gives no schedule, thus only a lower bound! construction of schedule: simple (page -4-) or via open shop (later)

10 Wrap around rule for problem P pmtn C max : define opt := max{max n j=1 p j, n j=1 p j/m} opt is a lower bound on the optimal value for problem P pmtn C max Construction of a schedule with C max = opt: fill the machines successively, schedule the jobs in any order and preempt a job if the time bound opt is met all jobs can be scheduled since opt n j=1 p j/m no job is scheduled at the same time on two machines since opt max n j=1 p j

11 Wrap around rule for problem P pmtn C max : Construction of a schedule with C max = opt: fill the machines successively, schedule the jobs in any order and preempt a job if the time bound opt is met all jobs can be scheduled since opt n j=1 p j/m no job is scheduled at the same time on two machines since opt max n j=1 p j Example: m = 3, n = 5, p = (3, 7, 5, 1, 4) M3 M2 M

12 Schedule construction via Open shop for P pmtn C max : given an optimal solution x of the LP, consider the following open shop instance n jobs, m machines and p ij := x ij p j solve for this instance O pmtn C max

13 Schedule construction via Open shop for P pmtn C max : given an optimal solution x of the LP, consider the open shop instance n jobs, m machines and p ij := x ij p j solve for this instance O pmtn C max Result: solution for problem P pmtn C max for O pmtn C max we show later that an optimal solution has value max{ n max j=1 m p ij, i=1 m max i=1 n p ij } j=1 and can be calculated in polynomial time Result: solution of O pmtn C max is optimal for P pmtn C max

14 Uniform machines: Q pmtn C max : m machines with speeds s 1,..., s m n jobs with processing times p 1,..., p n change LP!

15 Uniform machines: Q pmtn C max : m machines with speeds s 1,..., s m n jobs with processing times p 1,..., p n min s.t. C max n x ij p j /s i C max i = 1,..., m j=1 n x ij p j /s i C max j = 1,..., n i=1 m x ij = 1 j = 1,..., n i=1 x ij 0 i = 1,..., m; j = 1,..., n

16 Uniform machines: Q pmtn C max (cont.): since again no schedule is given, LP leads to lower bound for optimal value of Q pmtn C max, as for P pmtn C max we may solve an open shop instance corresponding to the optimal solution x of the LP with n jobs, m machines and p ij := x ij p j /s i this solution is an optimal schedule for Q pmtn C max

17 Unrelated machines: R pmtn C max : m machines n jobs with processing times p 1,..., p n speed s ij change LP!

18 Unrelated machines: R pmtn C max : m machines n jobs with processing times p 1,..., p n and given speeds s ij min s.t. C max n x ij p j /s ij C max i = 1,..., m j=1 n x ij p j /s ij C max j = 1,..., n i=1 m x ij = 1 j = 1,..., n i=1 x ij 0 i = 1,..., m; j = 1,..., n

19 Unrelated machines: R pmtn C max (cont.): same procedure as for Q pmtn C max! again no schedule is given, LP leads to lower bound for optimal value of R pmtn C max, for optimal solution x solve an corresponding open shop instance with n jobs, m machines and p ij := x ij p j /s ij this solution is an optimal schedule for R pmtn C max

20 Approximation methods for: P C max : list scheduling methods (based on priority rules) jobs are ordered in some sequence π always when a machine gets free, the next unscheduled job in π is assigned to that machine Theorem: List scheduling is a (2 1/m)-approximation for problem P C max for any given sequence π Proof on the board Holds also for P r j C max

21 Approximation methods for: P C max (cont.): consider special list LPT-rule (longest processing time first) is a natural candidate Theorem: The LPT-rule leads to a (4/3 1/3m)-approximation for problem P C max Proof on the board uses following result: Lemma: If an optimal schedule for problem P C max results in at most 2 jobs on any machine, then the LPT-rule is optimal Proof as Exercise the bound (4/3 1/3m) is tight (Exercise)

22 Parallel machine models: Total Completion Time Parallel machines: P C j : for m = 1, the SPT-rule is optimal (see Lecture 2) for m 2 a partition of the jobs is needed if a job j is scheduled as k-last job on a machine, this job contributes kp j to the objective value

23 Parallel machine models: Total Completion Time Parallel machines: P C j : for m = 1, the SPT-rule is optimal (see Lecture 2) for m 2 a partition of the jobs is needed if a job j is scheduled as k-last job on a machine, this job contributes kp j to the objective value we have m last positions where the processing time is weighted by 1, m second last positions where the processing time is weighted by 2, etc. use the n smallest weights for positioning the jobs

24 Parallel machine models: Total Completion Time Parallel machines: P C j : for m = 1, the SPT-rule is optimal (see Lecture 2) for m 2 a partition of the jobs is needed if a job j is scheduled as k-last job on a machine, this job contributes kp j to the objective value we have m last positions where the processing time is weighted by 1, m second last positions where the processing time is weighted by 2, etc. use the n smallest weights for positioning the jobs assign job with the ith largest processing time to ith smallest weight is optimal Result: SPT is also optimal for P C j

25 Parallel machine models: Total Completion Time Uniform machines: Q C j : if a job j is scheduled as k-last job on a machine M r, this job contributes kp j /s r = (k/s r )p j to the objective value; i.e. job j gets weight (k/s r ) for scheduling the n jobs on the m machines, we have weights { 1 s 1,..., 1, 2 2,...,,..., n,..., n } s m s 1 s m s 1 s m from these nm weights we select the n smallest weights and assign the ith largest job to the ith smallest weight leading to an optimal schedule

26 Parallel machine models: Total Completion Time Example uniform machines: Q C j : n = 6, p = (6, 9, 8, 12, 4, 2) m = 3, s = (3, 1, 4) possible weights: { 1 3, 1 1, 1 4, 2 3, 2 1, 2 4, 3 3, 3 1, 3 4, 4 3, 4 1, 4 4, 5 3, 5 1, 5 4, 6 3, 6 1, 6 4 } 6 smallest weights: { 1 3, 1 1, 1 4, 2 3, 2 1, 2 4, 3 3, 3 1, 3 4, 4 3, 4 1, 4 4, 5 3, 5 1, 5 4, 6 3, 6 1, 6 4 }

27 Parallel machine models: Total Completion Time Example uniform machines: Q C j : n = 6, p = (6, 9, 8, 12, 4, 2) m = 3, s = (3, 1, 4) 6 smallest weights: { 1 3, 1 1, 1 4, 2 3, 2 1, 2 4, 3 3, 3 1, 3 4, 4 3, 4 1, 4 4, 5 3, 5 1, 5 4, 6 3, 6 1, 6 4 } sorted list of weights: { 1 4, 1 3, 2 4, 2 3, 3 4, 4 4 } jobs sorted by decreasing processing times: (4, 2, 3, 1, 5, 6)

28 Parallel machine models: Total Completion Time Example uniform machines: Q C j : n = 6, p = (6, 9, 8, 12, 4, 2) m = 3, s = (3, 1, 4) sorted list of weights: { 1 4, 1 3, 2 4, 2 3, 3 4, 4 4 } jobs sorted by decreasing processing times: (4, 2, 3, 1, 5, 6) Schedule: M M2 M

29 Parallel machine models: Total Completion Time Unrelated machines: R C j : if a job j is scheduled as k-last job on a machine M r, this job contributes kp j /s rj to the objective value; since now the weight is also job-dependent, we cannot simply sort the weights assignment problem: n jobs nm machine positions (k, r) (k-last position on M r ) assigning job j to (k, r) has costs kp j /s rj find an assignment of minimal costs of all jobs to machine positions leads to optimal solution of R C j in polynomial time

30 Parallel machine models: Total Weighted Completion Time Parallel machines: P w j C j : Problem 1 w j C j is solvable via the WSPT-rule (Lecture 2) Problem P2 w j C j is...

31 Parallel machine models: Total Weighted Completion Time Parallel machines: P w j C j : Problem 1 w j C j is solvable via the WSPT-rule (Lecture 2) Problem P2 w j C j is already NP-hard, but Problem P2 w j C j is pseudopolynomial solvable Problem P w j C j is NP-hard in the strong sense Proof by reduction using 3-PARTITION as exercise Approximation:

32 Parallel machine models: Total Weighted Completion Time Parallel machines: P w j C j : Problem 1 w j C j is solvable via the WSPT-rule (Lecture 2) Problem P2 w j C j is already NP-hard, but Problem P2 w j C j is pseudopolynomial solvable Problem P w j C j is NP-hard in the strong sense Proof by reduction using 3-PARTITION as exercise Approximation: the WSPT-rule gives an 1 2 (1 + 2) approximation Proof is not given; uses fact that worst case examples have equal w j /p j ratios for all jobs

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