Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
97 tokens/sec
GPT-4o
53 tokens/sec
Gemini 2.5 Pro Pro
43 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
47 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A Subexponential Time Algorithm for Makespan Scheduling of Unit Jobs with Precedence Constraints (2312.03495v1)

Published 6 Dec 2023 in cs.DS

Abstract: In a classical scheduling problem, we are given a set of $n$ jobs of unit length along with precedence constraints, and the goal is to find a schedule of these jobs on $m$ identical machines that minimizes the makespan. Using the standard 3-field notation, it is known as $Pm|\text{prec}, p_j=1|C_{\max}$. Settling the complexity of $Pm|\text{prec}, p_j=1|C_{\max}$ even for $m=3$ machines is the last open problem from the book of Garey and Johnson [GJ79] for which both upper and lower bounds on the worst-case running times of exact algorithms solving them remain essentially unchanged since the publication of [GJ79]. We present an algorithm for this problem that runs in $(1+\frac{n}{m}){\mathcal{O}(\sqrt{nm})}$ time. This algorithm is subexponential when $m = o(n)$. In the regime of $m=\Theta(n)$ we show an algorithm that runs in$\mathcal{O}(1.997n)$ time. Before our work, even for $m=3$ machines there were no algorithms known that run in $\mathcal{O}((2-\varepsilon)n)$ time for some $\varepsilon > 0$.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (48)
  1. Fast FAST. In Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, volume 5555, pages 49–58. Springer, 2009.
  2. László Babai. Graph isomorphism in quasipolynomial time [extended abstract]. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 684–697. ACM, 2016.
  3. Subexponential-time algorithms for maximum independent set in Ptsubscript𝑃𝑡{P}_{t}italic_P start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT-free and broom-free graphs. Algorithmica, 81(2):421–438, 2019.
  4. Nikhil Bansal. Scheduling open problems: Old and new. MAPSP 2017, 2017.
  5. Andreas Björklund. Determinant sums for undirected hamiltonicity. SIAM Journal on Computing, 43(1):280–299, 2014.
  6. Fourier meets Möbius: fast subset convolution. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 67–74. ACM, 2007.
  7. A Discrete Subexponential Algorithm for Parity Games. In STACS 2003, pages 663–674. Springer Berlin Heidelberg, 2003.
  8. 𝖶𝖶\mathsf{W}sansserif_W[2]-hardness of precedence constrained k𝑘kitalic_k-processor scheduling. Operations Research Letters, 18(2):93–97, 1995.
  9. Parameterized Problems Complete for Nondeterministic FPT time and Logarithmic Space. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, pages 193–204, 2021.
  10. Deciding parity games in quasi-polynomial time. SIAM J. Comput., 51(2):17–152, 2022.
  11. On the optimality of exact and approximation algorithms for scheduling problems. J. Comput. Syst. Sci., 96:1–32, 2018.
  12. Optimal scheduling for two-processor systems. Acta informatica, 1(3):200–213, 1972.
  13. Scheduling partially ordered jobs faster than 2nsuperscript2𝑛2^{n}2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. Algorithmica, 68(3):692–714, 2014.
  14. A Simpler QPTAS for Scheduling Jobs with Precedence Constraints. In 30th Annual European Symposium on Algorithms, ESA 2022, volume 244 of LIPIcs, pages 40:1–40:11, 2022.
  15. A Framework for Exponential-Time-Hypothesis-Tight Algorithms and Lower Bounds in Geometric Intersection Graphs. SIAM J. Comput., 49(6):1291–1331, 2020.
  16. Scheduling precedence graphs of bounded height. Journal of Algorithms, 5(1):48–59, 1984.
  17. Bidimensionality. In Encyclopedia of Algorithms, pages 203–207. 2016.
  18. Subexponential Parameterized Algorithm for Minimum Fill-In. SIAM J. Comput., 42(6):2197–2216, 2013.
  19. Optimal sequencing of two equivalent processors. SIAM Journal on Applied Mathematics, 17(4):784–789, 1969.
  20. Harold N. Gabow. An almost-linear algorithm for two-processor scheduling. J. Assoc. Comput. Mach., 29(3):766–780, 1982.
  21. A linear-time algorithm for a special case of disjoint set union. Journal of computer and system sciences, 30(2):209–221, 1985.
  22. Computers and Intractability: A Guide to the Theory of 𝖭𝖯𝖭𝖯\mathsf{NP}sansserif_NP-Completeness. W. H. Freeman, 1979.
  23. Scheduling opposing forests. SIAM Journal on Algebraic Discrete Methods, 4(1):72–93, 1983.
  24. Shashwat Garg. Quasi-PTAS for Scheduling with Precedences using LP Hierarchies. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
  25. Independent Set on PksubscriptP𝑘\mathrm{P}_{k}roman_P start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT-Free Graphs in Quasi-Polynomial Time. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 613–624. IEEE, 2020.
  26. Tight Hardness Results for Training Depth-2 ReLU Networks. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1–22:14. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2021.
  27. Ronald L. Graham. Bounds for certain multiprocessing anomalies. The Bell System Technical Journal, 45(9):1563–1581, 1966.
  28. Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey. Annals of Discrete Mathematics, 5(2):287–326, 1979.
  29. A Dynamic Programming Approach to Sequencing Problems. Journal of the Society for Industrial and Applied mathematics, 10(1):196–210, 1962.
  30. Assembly-Line Balancing-Dynamic Programming with Precedence Constraints. Operations Research, 11(3):442–459, 1963.
  31. Te C. Hu. Parallel sequencing and assembly line problems. Operations research, 9(6):841–848, 1961.
  32. Precedence Scheduling with Unit Execution Time is Equivalent to Parametrized Biclique. In International Conference on Current Trends in Theory and Practice of Informatics, pages 329–343. Springer, 2016.
  33. A Deterministic Subexponential Algorithm for Solving Parity Games. SIAM J. Comput., 38(4):1519–1532, 2008.
  34. Critical-path planning and scheduling. In Papers presented at the December 1-3, 1959, eastern joint IRE-AIEE-ACM computer conference, pages 160–173, 1959.
  35. Sequencing and scheduling: Algorithms and complexity. Handbooks in operations research and management science, 4:445–522, 1993.
  36. Complexity of scheduling under precedence constraints. Operations Research, 26(1):22–35, 1978.
  37. A (1+ϵ)1italic-ϵ(1+\epsilon)( 1 + italic_ϵ )-Approximation for Makespan Scheduling with Precedence Constraints Using LP Hierarchies. SIAM J. Comput., 50(3), 2021.
  38. Shi Li. Towards PTAS for precedence constrained scheduling via combinatorial algorithms. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2991–3010. SIAM, 2021.
  39. Eugene M. Luks. Isomorphism of Graphs of Bounded Valence can be Tested in Polynomial Time. J. Comput. Syst. Sci., 25(1):42–65, 1982.
  40. Matthias Mnich and René van Bevern. Parameterized complexity of machine scheduling: 15 open problems. Computers & Operations Research, 2018.
  41. A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1682–1701. SIAM, 2021.
  42. Makespan scheduling of unit jobs with precedence constraints in o(1.995nn{}^{\mbox{n}}start_FLOATSUPERSCRIPT n end_FLOATSUPERSCRIPT) time. CoRR, abs/2208.02664, 2022.
  43. Scheduling interval-ordered tasks. SIAM Journal on Computing, 8(3):405–409, 1979.
  44. Polynomial time approximation algorithms for machine scheduling: Ten open problems. Journal of Scheduling, 2(5):203–213, 1999.
  45. Ravi Sethi. Scheduling graphs on two processors. SIAM J. Comput., 5(1):73–82, 1976.
  46. Ola Svensson. Conditional hardness of precedence constrained scheduling on identical machines. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 745–754, 2010.
  47. Moderate exponential-time algorithms for scheduling problems. 4OR, 20(4):533–566, 2022.
  48. Jeffrey D. Ullman. 𝖭𝖯𝖭𝖯\mathsf{NP}sansserif_NP-complete scheduling problems. Journal of Computer and System sciences, 10(3):384–393, 1975.
User Edit Pencil Streamline Icon: https://streamlinehq.com
Authors (3)
  1. Jesper Nederlof (46 papers)
  2. Céline M. F. Swennenhuis (9 papers)
  3. Karol Węgrzycki (37 papers)

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now