1. Completion Time Scheduling Notes from Hall, Schulz, Shmoys and Wein, Mathematics of Operations Research, Vol 22, 513-544, 1997

2. One LP formulation for 1||Σw j C j

3. Other ways to bound C j Smith’s rule: Scheduling jobs by w j /p j is guaranteed to be optimal wjwj pjpj

4. Think of all jobs as having w j = p j Smith’s rule: Any order is then equivalent since w j /p j = 1 for all jobs pjpj pjpj

6. 1|prec| Σw j C j LP: minimize Σw j C j Subject to C k ≥ C j + p k (if job j precedes job k) Let C’ i denote the LP optimal values Let C* i denote the true optimal values

7. Algorithm and Analysis Let C’ i denote the LP optimal values Let C* i denote the true optimal values Greedy Algorithm: –Solve LP for C’ i Solvable in poly time despite exponential size –Prioritize the jobs by C’ i values –Let G i be the resulting completion times Key result: G i ≤ 2C’ i ≤ 2C* i –From Lemma 2.1

8. 1|r j, prec| Σw j C j LP: minimize Σw j C j Subject to C j ≥ r j + p j (all jobs) C k ≥ C j + p k (if job j precedes job k) Let C’ i denote the LP optimal values Let C* i denote the true optimal values

9. Algorithm Let C’ i denote the LP optimal values Let C* i denote the true optimal values Greedy Algorithm: –Solve LP for C’ i Solvable in poly time despite exponential size –Prioritize the jobs by C’ i values –Let G i be the resulting completion times

10. Analysis Key result: G i ≤ 3C’ i ≤ 3C* i Fix j and define S = {1, …, j} G j ≤ r max (S) + p(S) C’ j ≥ r max (S) Thus G j ≤ C’ j + p(S) ≤ 3C’ j –From Lemma 2.1

11. Extending to parallel machines

13. P|r j | Σw j C j LP: minimize Σw j C j Subject to C j ≥ r j + p j Let C’ i denote the LP optimal values Let C* i denote the true optimal values

14. Algorithm Let C’ i denote the LP optimal values Let C* i denote the true optimal values Greedy Algorithm: –Solve LP for C’ i Solvable in poly time despite exponential size –Prioritize the jobs by C’ i values –Let G i be the resulting completion times

15. Analysis Key result: G i ≤ (4-1/m)C’ i ≤ 4C* i Fix j and define S = {1, …, j} G j ≤ r max (S) + 1/m p(S – {j}) + p j = r max (S) + 1/m p(S) + (1-1/m)p j

16. Preemption vs Nonpreemption Method for converting preemptive schedules into non-preemptive schedules –Effective for minimizing C j objectives Prioritize jobs by their preemptive completion times C j p –Generalization: When α of the job is complete List schedule these jobs nonpreemptively using this priority

17. 1 machine conversion Let C p i denote any preemptive values Let C n i denote the nonpreemptive values C n i ≤ 2 C p i … CPjCPj

18. 1|r j | ΣC j With preemption, we have an optimal solution, SRPT Nonpreemptive online: –Simulate SRPT and when a job is completed in SRPT, start it in the non-preemptive (or add it to the list to start)