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Fixed-Priority Schedulability of Sporadic Tasks on Uniprocessors is NP -hard Pontus Ekberg & Wang Yi Uppsala University RTSS 2017 Overview for coNP -complete Strongly Polynomial time utilization Arbitrary EDF c NP -hard coNP -complete

  1. Fixed-Priority Schedulability of Sporadic Tasks on Uniprocessors is NP -hard Pontus Ekberg & Wang Yi Uppsala University RTSS 2017

  2. Overview for coNP -complete Strongly Polynomial time utilization Arbitrary EDF c NP -hard coNP -complete Weakly time algorithm Exponential c for NP -complete Weakly c Strongly Utilization NP -hard c Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) for bounded by coNP -complete Weakly c for coNP -complete Weakly Polynomial time a constant c for Weakly Implicit utilization Weakly NP -hard Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. Arbitrary time algorithm FP ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines NP -complete Weakly NP -hard Weakly Pseudo-poly. RM priorities and ln for c Polynomial time time algorithm Pseudo-poly. a constant c bounded by Utilization time algorithm Exponential NP -hard Weakly NP -complete 2 ( d = p ) ( d ⩽ p )

  3. Overview Weakly Polynomial time utilization Arbitrary EDF c for NP -hard time algorithm coNP -complete Exponential c for NP -complete Weakly c for NP -hard Strongly Strongly Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 c for Weakly coNP -complete c for coNP -complete Weakly Polynomial time a constant c bounded by Utilization Weakly time algorithm Pseudo-poly. utilization NP -hard Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. Arbitrary Weakly FP ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines NP -complete Weakly NP -hard bounded by RM priorities and ln for c Polynomial time time algorithm Pseudo-poly. Weakly a constant c Utilization time algorithm Exponential NP -hard Weakly NP -complete 2 ( d = p ) ( d ⩽ p ) ∗ ∗ ∗ ∗ ( ∗ )

  4. Overview Weakly Polynomial time utilization Arbitrary EDF c for NP -hard time algorithm coNP -complete Exponential c for NP -complete Weakly c for NP -hard Strongly Strongly Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 Joseph and Pandya, 1986 c for Weakly coNP -complete c for coNP -complete Weakly Polynomial time a constant c bounded by Utilization Weakly 2 time algorithm utilization Pseudo-poly. Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. Arbitrary NP -complete FP ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Weakly NP -hard Weakly Utilization RM priorities Polynomial time time algorithm Pseudo-poly. NP -hard bounded by a constant c time algorithm Exponential NP -hard Weakly NP -complete Weakly ( d = p ) ( d ⩽ p ) ∗ ∗ ∗ † for c ⩽ ln 2 and ( ∗ ) ( † )

  5. Overview time algorithm utilization Arbitrary EDF c for NP -hard Weakly Exponential Strongly c for NP -complete Weakly c for NP -hard Weakly Polynomial time coNP -complete time algorithm Weakly Pontus Ekberg Lehoczky, 1990 Liu and Layland, 1973 Joseph and Pandya, 1986 c for coNP -complete c Strongly for coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete Implicit 2 Pseudo-poly. utilization Weakly Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. Arbitrary Weakly FP ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines NP -complete NP -hard NP -hard time algorithm RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly Utilization bounded by Exponential NP -hard Weakly NP -complete ( d = p ) ( d ⩽ p ) ∗ ∗ ‡ ∗ † ‡ for c ⩽ ln 2 and ( ∗ ) ( † ) ( ‡ )

  6. Overview for coNP -complete Strongly Polynomial time utilization Arbitrary EDF c NP -hard coNP -complete Weakly time algorithm Exponential c for NP -complete Weakly Strongly Utilization for c Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) for bounded by coNP -complete Weakly c for coNP -complete Weakly Polynomial time a constant c c NP -hard Implicit utilization NP -hard Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. Arbitrary Weakly FP ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Weakly NP -complete Weakly NP -hard time algorithm Pseudo-poly. RM priorities Polynomial time time algorithm Pseudo-poly. a constant c bounded by Utilization time algorithm Exponential NP -hard Weakly NP -complete Weakly 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and

  7. Overview Weakly utilization Arbitrary EDF c for NP -hard time algorithm Strongly Exponential c for NP -complete Weakly c for Polynomial time coNP -complete Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) Weakly Strongly coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete NP -hard Weakly time algorithm Arbitrary Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. utilization FP Weakly ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Pseudo-poly. NP -hard NP -complete Utilization RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly bounded by time algorithm Exponential NP -hard Weakly NP -complete Weakly NP -hard 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and for 0 < c < 1 for 0 < c < 1

  8. Overview Weakly utilization Arbitrary EDF c for NP -hard time algorithm Strongly Exponential c for NP -complete Weakly c for Polynomial time coNP -complete Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) Weakly Strongly coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete NP -hard Weakly time algorithm Arbitrary Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. utilization FP Weakly ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Pseudo-poly. NP -hard NP -complete Utilization RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly bounded by time algorithm Exponential NP -hard Weakly NP -complete Weakly NP -hard 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and for 0 < c < 1 for 0 < c < 1

  9. Overview Weakly utilization Arbitrary EDF c for NP -hard time algorithm Strongly Exponential c for NP -complete Weakly c for Polynomial time coNP -complete Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) Weakly Strongly coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete NP -hard Weakly time algorithm Arbitrary Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. utilization FP Weakly ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Pseudo-poly. NP -hard NP -complete Utilization RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly bounded by time algorithm Exponential NP -hard Weakly NP -complete Weakly NP -hard 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and for 0 < c < 1 for 0 < c < 1

  10. Overview Weakly utilization Arbitrary EDF c for NP -hard time algorithm Strongly Exponential c for NP -complete Weakly c for Polynomial time coNP -complete Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) Weakly Strongly coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete NP -hard Weakly time algorithm Arbitrary Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. utilization FP Weakly ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Pseudo-poly. NP -hard NP -complete Utilization RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly bounded by time algorithm Exponential NP -hard Weakly NP -complete Weakly NP -hard 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and for 0 < c < 1 for 0 < c < 1

  11. Overview Weakly utilization Arbitrary EDF c for NP -hard time algorithm Strongly Exponential c for NP -complete Weakly c for Polynomial time coNP -complete Implicit coNP -complete Pontus Ekberg Lehoczky, 1990 ( ) Liu and Layland, 1973 ( ) Joseph and Pandya, 1986 ( ) Weakly Strongly coNP -complete Weakly Polynomial time a constant c bounded by Utilization coNP -complete NP -hard Weakly time algorithm Arbitrary Weakly time algorithm Pseudo-poly. time algorithm Pseudo-poly. utilization FP Weakly ( d , p unrelated) deadlines Arbitrary deadlines Constrained deadlines Pseudo-poly. NP -hard NP -complete Utilization RM priorities Polynomial time time algorithm Pseudo-poly. a constant c Weakly bounded by time algorithm Exponential NP -hard Weakly NP -complete Weakly NP -hard 2 ( d = p ) ( d ⩽ p ) for c ⩽ ln 2 and for 0 < c < 1 for 0 < c < 1

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