Scheduling in the Random-Order Model Susanne Albers, Maximilian - - PowerPoint PPT Presentation

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scheduling in the random order model

Scheduling in the Random-Order Model Susanne Albers, Maximilian - - PowerPoint PPT Presentation

Sep 10, 2023 280 likes •998 views

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

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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

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Makespan Minimization

Task: Assign jobs to machines. 1 1 1 4 2 3.5 1.5 3.5

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

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Makespan Minimization

Task: Assign jobs to machines. 4 2 3.5 3.5 1 1 1 1.5

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 2

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Makespan Minimization

Task: Assign jobs to machines. Goal: Minimize the makespan. 4 2 3.5 3.5 1 1 1 1.5 makespan 5.5 load 4.5

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Makespan Minimization

Task: Assign jobs to machines. Goal: Minimize the makespan. 4 1 3.5 1 3.5 1 2 1.5 makespan 4.5

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2 1

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2 1 1

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2 1 1 2.5

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2 1 1 2.5 3

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 3

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Online Algorithm

Jobs are revealed one by one and assigned immediately. 1.5 2 1 1 2.5 3

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Competitive ratio

The worst ratio an adversary can cause. 1.5 2 1 1 2.5 3 1.5 2 1 1 2.5 3 versus

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Literature

Adversarial deterministic algorithms.

[Günther, Maurer, Megow and Wiese, 2013] and [Chen, Ye, Zhang, 2015] approximate the optimum

nline algorithm.

4/3 1.5 1.8 1.9 2

1.9201 [Fleischer, Wahl, 2000] 1.923 [Albers, 1999] 1.945 [Karger, Philipps, Torng, 1996] 1.986 [Bartal, Fiat, Karloff, Vohra, 1995] 2− 1

m−εm [Galambos, Woeginger, 1993]

2−1/m [Graham, 1966] [Rudin III, 2001] 1.885 [Gormley et. al., 2000] 1.853 [Albers, 1999] 1.852 [Bartal, Karloff, Rabani, 1994] 1.837 [Faigle, Kern, Turan, 1989] 1.707

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

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Literature

Adversarial deterministic algorithms. Adversarial randomized algorithms. 4/3 1.5 1.8 1.9 2

1.9201 [Fleischer, Wahl, 2000] [Rudin III, 2001] 1.885 1.916 [Albers, 2002] [Chen, Vliet, Woeginger, 1994] 1.5819 [Sgall, 1997] 1.5819

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Literature

Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT. (Bin Stretching) 1 4/3 1.5 1.8 1.9 2

1.5 [Böhm, Sgall, van Stee, Veselý, 2016] 1.53 [Gabay, Kotov, Brauner, 2013] 1.57 [Kellerer, Kotov, 2013] 1.625 [Azar, Regev, 1998] [Azar, Regev, 1998] 4/3

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

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Literature

Adversarial deterministic algorithms. Adversarial randomized algorithms. Deterministic algorithms which know OPT. (Bin Stretching) Deterministic algorithms which know the total processing volume. 1 4/3 1.5 1.8 1.9 2

1.585 [Kellerer, Kotov, Gabay, 2015] 1.6 [Cheng, Kellerer, Kotov, 2005] 1.725 [Angelli, Nagy, Speranza, 2004] 1.75 [Albers, Hellwig, 2012] [Albers, Hellwig, 2012] 1.585 [Angelli, Nagy, Speranza, 2004] 1.565 [Cheng, Kellerer, Kotov, 2005] 1.5

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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

1 4/3 1.5 1.8 1.9 2

[Albers, Hellwig, 2013] [Dohrau, 2015] I made some corrections here.

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

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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 4/3 1.5 1.8 1.9 2

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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 4/3 1.5 1.8 1.9 2

1.25 [Cheng, Kellerer, Kotov, 2012] 4/3 [Graham, 1969] [Seiden et. al., 2000] 1.1805

Maximilian Janke (TUM) | OLAWA 2020 | Scheduling in the Random-Order Model 4

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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. 4/3 1.5 1.8 1.9 2

1.9201 [Fleischer, Wahl, 2000] 1.923 [Albers, 1999] 1.945 [Karger, Philipps, Torng, 1996] 1.986 [Bartal, Fiat, Karloff, Vohra, 1995] 2− 1

m−εm [Galambos, Woeginger, 1993]

2−1/m [Graham, 1966] [Rudin III, 2001] 1.885 [Gormley et. al., 2000] 1.853 [Albers, 1999] 1.852 [Bartal, Karloff, Rabani, 1994] 1.837 [Faigle, Kern, Turan, 1989] 1.707

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Is worst-case analysis too pessimistic?

(Lower bound of [Albers, 1999] for m = 40 machines.)

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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)

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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.

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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.

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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?

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Random-Order Analysis

(Random permutation of [Albers, 1999] for m = 40 machines.)

Adversary chooses job set, order is uniformly random.

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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.

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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.

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Is Random-Order Analysis too optimistic?

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Is Random-Order Analysis too optimistic?

We are nearly c-competitive iff we are (c +om(1))-competitive with probability 1−om(1) after random permutation and have a constant competitive ratio on worst-case sequences.

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Is Random-Order Analysis too optimistic?

We are nearly c-competitive iff we are (c +om(1))-competitive with probability 1−om(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 → ∞.

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The Random-Order model

Scheduling for the random-order model is not well researched yet:

[Osborn and Torng, 2008] show Greedy remains 2-competitive.

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The Random-Order model

Scheduling for the random-order model is not well researched yet:

[Osborn and Torng, 2008] show Greedy remains 2-competitive.
[Göbel, Kesselheim and Tönnis, 2015] and [Molinaro, 2017] analyze

different scheduling problems.

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Our results

Our new algorithm is nearly 1.8478-competitive. We also provide lower bounds.

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Our results

Our new algorithm is nearly 1.8478-competitive. decreasing order random-order adversarial 1 4/3 1.5 1.8 1.9 2

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Profit from random-order arrival.

The Load Lemma. Identifying time measures.

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Profit from random-order arrival.

The Load Lemma. Identifying time measures.
Large Job Magic. Large jobs for free.

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Profit from random-order arrival.

The Load Lemma. Identifying time measures.
Large Job Magic. Large jobs for free.
Oracle-like properties. A taste of semi-online scheduling.

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Profit from random-order arrival.

The Load Lemma. Identifying time measures.
Large Job Magic. Large jobs for free.
Oracle-like properties. A taste of semi-online scheduling.
???

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Profit from random-order arrival.

The Load Lemma. Identifying time measures.
Large Job Magic. Large jobs for free.
Oracle-like properties. A taste of semi-online scheduling.
???
Profit. A nearly 1.8478-competitive algorithm.

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Different time measures.

When have we seen half the sequence?

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Different time measures.

When have we seen half the sequence? The naive measure:

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Different time measures.

When have we seen half the sequence? The naive measure versus the load measure.

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Different time measures.

When have we seen half the sequence? The naive measure versus the load measure.

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Different time measures.

When have we seen half the sequence? The naive measure versus the load measure.

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Different time measures.

The naive measure versus the load measure.

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Different time measures.

How does random-order arrival impact the measures?

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Different time measures.

Load measure and naive measure agree

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Different time measures.

Load measure and naive measure agree

for m large high probability non-simple inputs margin of error

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How to foil reordering arguments.

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How to foil reordering arguments.

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How to foil reordering arguments.

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Large Job Magic: Large jobs for free

How to complete the sequence correctly?

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Large Job Magic: Large jobs for free

How to complete the sequence correctly?

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Large Job Magic: Large jobs for free

No high concentration of large jobs.

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Large Job Magic: Large jobs for free

We call it Large Job Magic.

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Oracle-like properties

A difficult sequence has large jobs close to the end. versus

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Oracle-like properties

A difficult sequence has large jobs close to the end. versus [Osborn and Torng, 2008] Highly probable if there are many large jobs.

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Oracle-like properties

A difficult sequence has large jobs close to the end. pn Highly probable if there are many large jobs. Many large jobs work as an oracle for OPT.

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Oracle-like properties

A difficult sequence has large jobs close to the end. pn Highly probable if there are many large jobs. Many large jobs work as an oracle for OPT. A competitive ratio of about 1.89 in expectation.

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Oracle-like properties

A difficult sequence has large jobs close to the end. pn Highly probable if there are many large jobs. Many large jobs work as an oracle for OPT. The difficult part is obtaining such improvement with high probability

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Oracle-like properties

Idea: Make the algorithm robust towards few large jobs.

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Oracle-like properties

Idea: Make the algorithm robust towards few large jobs. Problem: This already requires the ’oracle’ .

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Oracle-like properties

Idea: Make the algorithm robust towards few large jobs. Problem: This already requires the ’oracle’ . Till ’oracle’ is huge. ’oracle’ Omid Pend

m+1

Oend

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Take-away

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Load Lemma:

Relate algorithmic and probabilistic properties.

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Load Lemma:

Relate algorithmic and probabilistic properties.

Large Job Magic: Get large jobs for free.

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Load Lemma:

Relate algorithmic and probabilistic properties.

Large Job Magic: Get large jobs for free. Oracle-like properties: Predict the future.

Till Ph is huge.

Ph Omid Pend

m+1

Oend

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Thank you!

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