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Scheduling in the Random-Order Model Susanne Albers, Maximilian Janke Technical University of Munich Department of Computer Science Chair of Algorithms and Complexity Makespan Minimization Task: Assign jobs to machines . 4 3.5 3.5 2 1.5 1

  1. Scheduling in the Random-Order Model Susanne Albers, Maximilian Janke Technical University of Munich Department of Computer Science Chair of Algorithms and Complexity

  2. Makespan Minimization Task: Assign jobs to machines . 4 3.5 3.5 2 1.5 1 1 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

  3. Makespan Minimization Task: Assign jobs to machines . 3.5 1.5 1 4 3.5 1 2 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

  4. Makespan Minimization Task: Assign jobs to machines . Goal: Minimize the makespan . 3.5 1.5 makespan 5.5 1 load 4.5 4 3.5 1 2 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

  5. Makespan Minimization Task: Assign jobs to machines . Goal: Minimize the makespan . 1.5 3.5 3.5 makespan 4.5 4 2 1 1 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

  6. Online Algorithm Jobs are revealed one by one and assigned immediately. 1.5 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  7. Online Algorithm Jobs are revealed one by one and assigned immediately. 2 1.5 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  8. Online Algorithm Jobs are revealed one by one and assigned immediately. 1 2 1.5 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  9. Online Algorithm Jobs are revealed one by one and assigned immediately. 1 2 1.5 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  10. Online Algorithm Jobs are revealed one by one and assigned immediately. 2.5 1 2 1.5 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  11. Online Algorithm Jobs are revealed one by one and assigned immediately. 3 1 2.5 2 1.5 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  12. Online Algorithm Jobs are revealed one by one and assigned immediately. 3 1 2.5 2 1.5 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  13. Competitive ratio The worst ratio an adversary can cause. 3 1 1 3 1 2.5 2.5 versus 2 2 1.5 1.5 1 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

  14. Literature Adversarial deterministic algorithms. [Günther, Maurer, Megow and Wiese, 2013] and [Chen, Ye, Zhang, 2015] approximate the optimum online algorithm. [Rudin III, 2001] 1 . 885 1 . 9201 [Fleischer, Wahl, 2000] [Gormley et. al., 2000] 1 . 853 1 . 923 [Albers, 1999] [Albers, 1999] 1 . 852 1 . 945 [Karger, Philipps, Torng, 1996] [Bartal, Karloff, Rabani, 1994] 1 . 837 1 . 986 [Bartal, Fiat, Karloff, Vohra, 1995] 2 − 1 [Faigle, Kern, Turan, 1989] 1 . 707 m − ε m [Galambos, Woeginger, 1993] 2 − 1 / m [Graham, 1966] 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  15. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. [Rudin III, 2001] 1 . 885 1 . 9201 [Fleischer, Wahl, 2000] [Chen, Vliet, Woeginger, 1994] 1 . 5819 [Sgall, 1997] 1 . 5819 1 . 916 [Albers, 2002] 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  16. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) 1 . 5 [Böhm, Sgall, van Stee, Veselý, 2016] 1 . 53 [Gabay, Kotov, Brauner, 2013] [Azar, Regev, 1998] 4 / 3 1 . 57 [Kellerer, Kotov, 2013] 1 . 625 [Azar, Regev, 1998] 1 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  17. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) Deterministic algorithms which know the total processing volume. [Albers, Hellwig, 2012] 1 . 585 1 . 585 [Kellerer, Kotov, Gabay, 2015] [Angelli, Nagy, Speranza, 2004] 1 . 565 1 . 6 [Cheng, Kellerer, Kotov, 2005] [Cheng, Kellerer, Kotov, 2005] 1 . 5 1 . 725 [Angelli, Nagy, Speranza, 2004] 1 . 75 [Albers, Hellwig, 2012] 1 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  18. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) Deterministic algorithms which know the total processing volume. Deterministic algorithms which advice. constant in input length constant in m [Albers, Hellwig, 2013] [Dohrau, 2015] 1 1 . 5 1 . 8 1 . 9 2 4 / 3 I made some corrections here. Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  19. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) Deterministic algorithms which know the total processing volume. Deterministic algorithms which advice. Buffer Reordering. 1 . 4659 [Englert, Özmen, Westermann, 2008] 1 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  20. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) Deterministic algorithms which know the total processing volume. Deterministic algorithms which advice. Buffer Reordering. Job processing times ordered decreasingly. 1 . 25 [Cheng, Kellerer, Kotov, 2012] [Seiden et. al., 2000] 1 . 1805 4 / 3 [Graham, 1969] 1 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  21. Literature Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT . (Bin Stretching) Deterministic algorithms which know the total processing volume. Deterministic algorithms which advice. Buffer Reordering. [Rudin III, 2001] 1 . 885 1 . 9201 [Fleischer, Wahl, 2000] [Gormley et. al., 2000] 1 . 853 1 . 923 [Albers, 1999] [Albers, 1999] 1 . 852 1 . 945 [Karger, Philipps, Torng, 1996] [Bartal, Karloff, Rabani, 1994] 1 . 837 1 . 986 [Bartal, Fiat, Karloff, Vohra, 1995] 2 − 1 [Faigle, Kern, Turan, 1989] 1 . 707 m − ε m [Galambos, Woeginger, 1993] 2 − 1 / m [Graham, 1966] 1 . 5 1 . 8 1 . 9 2 4 / 3 Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

  22. Is worst-case analysis too pessimistic? (Lower bound of [Albers, 1999] for m = 40 machines.) Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 5

  23. Is worst-case analysis too pessimistic? (Lower bound of [Albers, 1999] for m = 40 machines.) The argument of Albers does not hold anymore if we • Delete any job. (The lower bound would be 1 . 714) Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 5

  24. Is worst-case analysis too pessimistic? (Lower bound of [Albers, 1999] for m = 40 machines.) The argument of Albers does not hold anymore if we • Delete any job. • Swap (non-identical) jobs. Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 5

  25. Is worst-case analysis too pessimistic? (Lower bound of [Albers, 1999] for m = 40 machines.) The argument of Albers does not hold anymore if we • Delete any job. • Swap (non-identical) jobs. • Change a job size significantly. Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 5

  26. Is worst-case analysis too pessimistic? (Lower bound of [Albers, 1999] for m = 40 machines.) The argument of Albers does not hold anymore if we • Swap (non-identical) jobs. How important is job order? Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 5

  27. Random-Order Analysis (Random permutation of [Albers, 1999] for m = 40 machines.) Adversary chooses job set , order is uniformly random . Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 6

  28. Random-Order Analysis (Random permutation of [Albers, 1999] for m = 40 machines.) Adversary chooses job set , order is uniformly random . Expected makespan of A versus optimum makespan . Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 7

  29. Random-Order Analysis (Random permutation of [Albers, 1999] for m = 40 machines.) Adversary chooses job set , order is uniformly random . Expected makespan of A versus optimum makespan . Competitive ratio: Worst ratio the adversary may cause. Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 8

  30. Is Random-Order Analysis too optimistic? Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 9

  31. Is Random-Order Analysis too optimistic? We are nearly c-competitive iff we are ( c + o m ( 1 )) -competitive with probability 1 − o m ( 1 ) after random permutation and have a constant competitive ratio on worst-case sequences. Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 10

  32. Is Random-Order Analysis too optimistic? We are nearly c-competitive iff we are ( c + o m ( 1 )) -competitive with probability 1 − o m ( 1 ) after random permutation and have a constant competitive ratio on worst-case sequences. A nearly c-competitive online algorithm is c-competitive in the random-order model for m → ∞ . Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 11

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