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

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 constrained deadlines. Instances Is T feasible on a preemptive uniprocessor? Qvestion Pontus Ekberg Feasibility is coNP-complete Under Bounded Utilization 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

• In coNP • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Conjectured poly. time for all c • Weakly coNP -hard for all c • Exp. time algorithm exists • Pseudo-poly. time algorithm if c • In coNP Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T [Eisenbrand & Rothvoß, SODA’10] An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? T e d p lcm p T 3 ℓ

• In coNP • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Conjectured poly. time for all c • Weakly coNP -hard for all c • Exp. time algorithm exists • Pseudo-poly. time algorithm if c • In coNP Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] [Eisenbrand & Rothvoß, SODA’10] Feasibility U T p.p. c p.p. if U T ? 3 ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • In coNP • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] [Eisenbrand & Rothvoß, SODA’10] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. c p.p. if U T ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T }

• Conjectured poly. time for all c • Weakly coNP -hard for all c • Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Pseudo-poly. time algorithm if c • In coNP An Algorithm for Feasibility [Baruah et al., 1990] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg c ) c -Feasibility (U T [Eisenbrand & Rothvoß, SODA’10] Feasibility U T p.p. ? 3 • Exp. time algorithm exists • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1

• Weakly coNP -hard • Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Conjectured poly. time for all c • Weakly coNP -hard for all c [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg An Algorithm for Feasibility [Baruah et al., 1990] Feasibility U T p.p. ? 3 • Exp. time algorithm exists • In coNP c -Feasibility (U ( T ) ⩽ c ) • Pseudo-poly. time algorithm if c < 1 • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1

• Conjectured pseudo-poly. time • Strongly coNP -hard [ECRTS’15] • Conjectured poly. time for all c • Weakly coNP -hard for all c An Algorithm for Feasibility [Baruah et al., 1990] [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg 3 Feasibility U T p.p. ? • Exp. time algorithm exists • In coNP • Weakly coNP -hard c -Feasibility (U ( T ) ⩽ c ) • Pseudo-poly. time algorithm if c < 1 • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1

• Strongly coNP -hard [ECRTS’15] • Conjectured poly. time for all c • Weakly coNP -hard for all c An Algorithm for Feasibility [Baruah et al., 1990] [Eisenbrand & Rothvoß, SODA’10] Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg 3 Feasibility U T p.p. ? • Exp. time algorithm exists • In coNP • Weakly coNP -hard • Conjectured pseudo-poly. time c -Feasibility (U ( T ) ⩽ c ) • Pseudo-poly. time algorithm if c < 1 • In coNP ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1

• Strongly coNP -hard [ECRTS’15] • Weakly coNP -hard for all c An Algorithm for Feasibility [Baruah et al., 1990] Feasibility Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg [Eisenbrand & Rothvoß, SODA’10] 3 U T p.p. ? • Exp. time algorithm exists • In coNP • Weakly coNP -hard • Conjectured pseudo-poly. time c -Feasibility (U ( T ) ⩽ c ) • Pseudo-poly. time algorithm if c < 1 • In coNP • Conjectured poly. time for all c < 1 ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1

• Weakly coNP -hard for all c An Algorithm for Feasibility [Baruah et al., 1990] Feasibility Feasibility is coNP-complete Under Bounded Utilization Pontus Ekberg [Eisenbrand & Rothvoß, SODA’10] 3 U T p.p. ? • Exp. time algorithm exists • In coNP • Weakly coNP -hard • 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 ℓ P ( T ) = lcm { p | ( e , d , p ) ∈ T } p.p. if U ( T ) ⩽ c < 1