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Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly coNP-complete Pontus Ekberg & Wang Yi Uppsala University ECRTS 2015 Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP

  1. Uniprocessor Feasibility of Sporadic Tasks with Constrained Deadlines is Strongly coNP-complete Pontus Ekberg & Wang Yi Uppsala University ECRTS 2015

  2. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  3. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  4. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  5. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  6. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  7. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  8. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  9. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  10. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  11. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  12. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  13. Context on the Uniprocessor Feasibility Problem ? coNP -complete ? Strongly coNP -complete ? Pseudo-poly. solution exists 1980 ? 1990 2000 2010 Leung & Merrill Baruah et al. Eisenbrand & Rothvoß Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete (Weakly) coNP -complete General case ? Utilization bounded by a Asynchronous periodic Synchronous periodic (or sporadic) (Weakly) Strongly coNP -hard ? Strongly coNP -complete ? (Weakly) coNP -hard ? 2 constant c < 1

  14. How? SCP SCP Strongly NP -complete (Baruah et al., 1990) Pseudo-polynomial transformation in- Feasibility in- Feasibility Strongly NP -hard Strongly coNP -hard Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3 ∝

  15. How? SCP SCP Strongly NP -complete (Baruah et al., 1990) Pseudo-polynomial transformation in- Feasibility in- Feasibility Strongly NP -hard Strongly coNP -hard Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3 ∝

  16. How? SCP SCP Strongly NP -complete (Baruah et al., 1990) Pseudo-polynomial transformation in- Feasibility in- Feasibility Strongly NP -hard Strongly coNP -hard Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3 ∝

  17. How? SCP SCP Strongly NP -complete (Baruah et al., 1990) Pseudo-polynomial transformation in- Feasibility in- Feasibility Strongly NP -hard Strongly coNP -hard Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3 ∝

  18. How? SCP SCP Strongly NP -complete (Baruah et al., 1990) Pseudo-polynomial transformation in- Feasibility in- Feasibility Strongly NP -hard Strongly coNP -hard Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 3 ∝

  19. The Simultaneous Congruences Problem (SCP) An SCP instance is given by a pair A k . Example: A k A ? Yes A ? No Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4

  20. The Simultaneous Congruences Problem (SCP) Example: A k A ? Yes A ? No Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4 An SCP instance is given by a pair ( A , k ) .

  21. The Simultaneous Congruences Problem (SCP) Example: k A ? Yes A ? No Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4 An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) }

  22. The Simultaneous Congruences Problem (SCP) Example: A ? Yes A ? No Pontus Ekberg Sporadic Feasibility is Strongly coNP-complete 4 An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) } k = 2

  23. The Simultaneous Congruences Problem (SCP) A Sporadic Feasibility is Strongly coNP-complete Pontus Ekberg No ? A Yes ? 4 Example: An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) } k = 2 0 2 4 6 8 10 12 14 16 18 20

  24. The Simultaneous Congruences Problem (SCP) A Sporadic Feasibility is Strongly coNP-complete Pontus Ekberg No ? A Yes ? 4 Example: An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) } k = 2 (2 , 4) 0 2 4 6 8 10 12 14 16 18 20

  25. The Simultaneous Congruences Problem (SCP) A Sporadic Feasibility is Strongly coNP-complete Pontus Ekberg No ? A Yes ? 4 Example: An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) } k = 2 (2 , 4) (4 , 6) 0 2 4 6 8 10 12 14 16 18 20

  26. The Simultaneous Congruences Problem (SCP) A Sporadic Feasibility is Strongly coNP-complete Pontus Ekberg No ? A Yes ? 4 Example: An SCP instance is given by a pair ( A , k ) . A = { (2 , 4) , (4 , 6) , (3 , 8) , (0 , 3) } k = 2 (2 , 4) (4 , 6) (3 , 8) 0 2 4 6 8 10 12 14 16 18 20

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