Uniprocessor Feasibility of Sporadic Tasks Remains coNP-complete - - PowerPoint PPT Presentation

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Uniprocessor Feasibility of Sporadic Tasks Remains coNP-complete - - PowerPoint PPT Presentation

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Uniprocessor Feasibility of Sporadic Tasks Remains coNP-complete Under Bounded Utilization Pontus Ekberg & Wang Yi Uppsala University RTSS, December 2015 The General Setting Task set T of sporadic (or synchronous periodic) tasks with

slide-1
SLIDE 1

Uniprocessor Feasibility of Sporadic Tasks Remains coNP-complete Under Bounded Utilization

Pontus Ekberg & Wang Yi

Uppsala University

RTSS, December 2015

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SLIDE 2

The General Setting

Task set T of sporadic (or synchronous periodic) tasks with constrained deadlines. Instances Is T feasible on a preemptive uniprocessor? Qvestion

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 2

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SLIDE 3

The General Setting

Task set T of sporadic (or synchronous periodic) tasks with constrained deadlines. Instances Is T feasible on a preemptive uniprocessor? Qvestion

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 2

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SLIDE 4

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-5
SLIDE 5

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-6
SLIDE 6

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-7
SLIDE 7

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-8
SLIDE 8

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-9
SLIDE 9

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-10
SLIDE 10

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-11
SLIDE 11

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-12
SLIDE 12

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-13
SLIDE 13

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

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SLIDE 14

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U T

c)

  • Pseudo-poly. time algorithm if c
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

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SLIDE 15

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

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SLIDE 16

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-17
SLIDE 17

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-18
SLIDE 18

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

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SLIDE 19

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

P(T) = lcm{p | (e, d, p) ∈ T}

?

p.p. if U(T) ⩽ c < 1 p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-20
SLIDE 20

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U T

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-21
SLIDE 21

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T)

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

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SLIDE 22

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T) 0.9

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-23
SLIDE 23

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T) 0.9 0.99

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-24
SLIDE 24

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T) 0.9 0.99 0.5

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-25
SLIDE 25

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T) 0.9 0.99 0.5 0.05

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-26
SLIDE 26

An Algorithm for Feasibility [Baruah et al., 1990]

ℓ

T lcm p e d p T

?

p.p. if U T c p.p. U(T) 0.9 0.99 0.5 0.05

Feasibility

  • Exp. time algorithm exists
  • In coNP
  • Weakly coNP-hard

[Eisenbrand & Rothvoß, SODA’10]

  • Conjectured pseudo-poly. time
  • Strongly coNP-hard [ECRTS’15]

c-Feasibility (U(T) ⩽ c)

  • Pseudo-poly. time algorithm if c < 1
  • In coNP
  • Conjectured poly. time for all c < 1
  • Weakly coNP-hard for all c > 0

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 3

slide-27
SLIDE 27

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible

T

c is feasible 2 U T c

c

3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-28
SLIDE 28

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible

T

c is feasible 2 U T c

c

3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-29
SLIDE 29

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible

T

c is feasible 2 U T c

c

3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-30
SLIDE 30

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible

T

c is feasible 2 U T c

c

3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-31
SLIDE 31

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible

T

c is feasible 2 U T c

c

3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-32
SLIDE 32

Proving Hardness for c-Feasibility

Feasibility c-Feasibility

T T

c

1 T is feasible ⇔ T c is feasible 2 U(T c) ⩽ c 3 T c is computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 4

slide-33
SLIDE 33

Feasibility ∝ c-Feasibility, Step 1

ℓ

T T T

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-34
SLIDE 34

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-35
SLIDE 35

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-36
SLIDE 36

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-37
SLIDE 37

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-38
SLIDE 38

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

+ =

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-39
SLIDE 39

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-40
SLIDE 40

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-41
SLIDE 41

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-42
SLIDE 42

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved ✓ Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-43
SLIDE 43

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved ✓ Utilization ⩽ c ✗ Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-44
SLIDE 44

Feasibility ∝ c-Feasibility, Step 1

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved ✓ Utilization ⩽ c ✗ Computed in poly. time ✓ Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 5

slide-45
SLIDE 45

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-46
SLIDE 46

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-47
SLIDE 47

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-48
SLIDE 48

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization ⩽ c Computed in poly. time Feasibility preserved ✗ Utilization c Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-49
SLIDE 49

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved ✗ Utilization ⩽ c ✓ Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-50
SLIDE 50

Feasibility ∝ c-Feasibility, Step 2

ℓ

P(T) 2P(T) 3P(T)

Feasibility preserved Utilization c Computed in poly. time Feasibility preserved ✗ Utilization ⩽ c ✓ Computed in poly. time ✓ Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 6

slide-51
SLIDE 51

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-52
SLIDE 52

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-53
SLIDE 53

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-54
SLIDE 54

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-55
SLIDE 55

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-56
SLIDE 56

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization ⩽ c ✗ Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-57
SLIDE 57

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization ⩽ c ✗ Utilization c Computed in poly. time Computed in poly. time ✓ Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-58
SLIDE 58

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-59
SLIDE 59

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-60
SLIDE 60

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-61
SLIDE 61

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-62
SLIDE 62

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-63
SLIDE 63

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-64
SLIDE 64

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization c Utilization ⩽ c ✓ Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-65
SLIDE 65

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization c Utilization ⩽ c ✓ Computed in poly. time Computed in poly. time Computed in poly. time ✗ Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-66
SLIDE 66

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-67
SLIDE 67

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-68
SLIDE 68

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-69
SLIDE 69

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1 [3]P(T)

T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-70
SLIDE 70

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1 [3]P(T)

T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-71
SLIDE 71

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T T T

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-72
SLIDE 72

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T

[2]P(T) [3]P(T)

Feasibility preserved Feasibility preserved Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-73
SLIDE 73

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T

[2]P(T) [3]P(T)

Feasibility preserved Feasibility preserved ✓ Utilization ⩽ c Utilization c Utilization c Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-74
SLIDE 74

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T

[2]P(T) [3]P(T)

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization c Utilization ⩽ c ✓ Computed in poly. time Computed in poly. time Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-75
SLIDE 75

Feasibility ∝ c-Feasibility, Step 3

ℓ

P(T) 2P(T) 3P(T)

1

T

[2]P(T) [3]P(T)

Feasibility preserved Feasibility preserved ✓ Utilization c Utilization c Utilization ⩽ c ✓ Computed in poly. time Computed in poly. time ✓ Computed in poly. time Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 7

slide-76
SLIDE 76

Feasibility ∝ c-Feasibility, With log-scale Glasses

log

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-77
SLIDE 77

Feasibility ∝ c-Feasibility, With log-scale Glasses

ℓ log

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-78
SLIDE 78

Feasibility ∝ c-Feasibility, With log-scale Glasses

ℓ log

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-79
SLIDE 79

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-80
SLIDE 80

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-81
SLIDE 81

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-82
SLIDE 82

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-83
SLIDE 83

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1

T T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-84
SLIDE 84

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T)

T T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-85
SLIDE 85

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T) [3]P(T)

T T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-86
SLIDE 86

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T) [3]P(T) [4]P(T)

T T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-87
SLIDE 87

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T) [3]P(T) [4]P(T) [5]P(T)

T T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-88
SLIDE 88

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T) [3]P(T) [4]P(T) [5]P(T) [6]P(T)

T

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-89
SLIDE 89

Feasibility ∝ c-Feasibility, With log-scale Glasses

log(ℓ)

1 [2]P(T) [3]P(T) [4]P(T) [5]P(T) [6]P(T) [7]P(T) Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 8

slide-90
SLIDE 90

When is Feasibility Decidable in Poly. Time?

1 If deadlines are implicit. [Liu & Layland, 1973] 2 If deadlines are constrained and periods are harmonic.

[Bonifaci et al., 2013]

3 If U(T) ⩽ c < 1 and

max period min period ⩽ q(n).

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 9

slide-91
SLIDE 91

When is Feasibility Decidable in Poly. Time?

1 If deadlines are implicit. [Liu & Layland, 1973] 2 If deadlines are constrained and periods are harmonic.

[Bonifaci et al., 2013]

3 If U(T) ⩽ c < 1 and

max period min period ⩽ q(n).

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 9

slide-92
SLIDE 92

When is Feasibility Decidable in Poly. Time?

1 If deadlines are implicit. [Liu & Layland, 1973] 2 If deadlines are constrained and periods are harmonic.

[Bonifaci et al., 2013]

3 If U(T) ⩽ c < 1 and

max period min period ⩽ q(n).

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 9

slide-93
SLIDE 93

Conclusion

General case Utilization bounded by a constant c, 0 < c < 1 Asynchronous periodic Synchronous periodic (or sporadic) Strongly coNP-complete Strongly coNP-complete Strongly coNP-complete Weakly coNP-complete

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 10

slide-94
SLIDE 94

∀Tiank you! ⋄ ∃Qvestions?

Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 11