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Multi-granularity hybrid parallel network simplex algorithm for minimum-cost flow problems

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Abstract

Minimum-cost flow problems widely exist in graph theory, computer science, information science, and transportation science. The network simplex algorithm is a fast and frequently used method for solving minimum-cost flow problems. However, the conventional sequential algorithms cannot satisfy the requirement of high-computational efficiency for large-scale networks. Parallel computing has resulted in numerous significant advances in science and technology over the past decades and is potential to develop an effective means to solve the computational bottleneck problem of large-scale networks. This paper first analyzes the parallelizability of network simplex algorithm and then presents a multi-granularity parallel network simplex algorithm (MPNSA) with fine- and coarse-granularity parallel strategies, which are suitable for shared- and distributed-memory parallel applications, respectively. MPNSA is achieved by message-passing interface, open multiprocessing, and compute unified device architecture, so that it can be compatible with different high-performance computing platforms. Experimental results demonstrated that MPNSA has very great accelerating effects and the maximum speedup reaches 18.7.

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References

  1. Rao L, Liu X, Xie L, Liu W (2010) Minimizing electricity cost: optimization of distributed internet data centers in a multi-electricity-market environment. In: INFOCOM, 2010 Proceedings IEEE. IEEE, pp 1–9

  2. Zhao S, Sun X, Chen J, Duan Z, Zhang Y, Zhang Y (2018) Relational granulation method based on quotient space theory for maximum flow problem. Inf Sci. https://doi.org/10.1016/j.ins.2018.12.009

    Article  Google Scholar 

  3. Han S, Peng Z, Wang S (2014) The maximum flow problem of uncertain network. Inf Sci 265:167–175

    Article  MathSciNet  MATH  Google Scholar 

  4. Chiaraviglio L, Mellia M, Neri F (2012) Minimizing ISP network energy cost: formulation and solutions. IEEE/ACM Trans Netw 20(2):463–476

    Article  Google Scholar 

  5. Basten RJI, Van der Heijden MC, Schutten JMJ (2011) Practical extensions to a minimum cost flow model for level of repair analysis. Eur J Oper Res 211(2):333–342

    Article  MathSciNet  Google Scholar 

  6. Jin M, Granda-Marulanda NA, Down I (2014) The impact of carbon policies on supply chain design and logistics of a major retailer. J Clean Prod 85:453–461

    Article  Google Scholar 

  7. Xie F, Jia Y, Jia R (2012) Algorithm for minimum cost maximum flow in transportation network. J Converg Inf Technol 7(7):165–173

    Google Scholar 

  8. Ahuja RK, Magnanti TL, Orlin JB (1993) Network flows: theory, algorithms and applications. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  9. Edmonds J, Karp RM (1972) Theoretical improvements in algorithmic efficiency for network flow problems. J ACM 19(2):248–264

    Article  MATH  Google Scholar 

  10. Goldberg AV, Tarjan RE (1989) Finding minimum-cost circulations by canceling negative cycles. J ACM 36(4):873–886

    Article  MathSciNet  MATH  Google Scholar 

  11. Sokkalingam PT, Ahuja RK, Orlin JB (2000) New polynomial time cycle-canceling algorithms for minimum-cost flows. Networks 36:53–63

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldberg A, Tarjan R (1987) Solving minimum cost flow problem by successive approximation. In: Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing. ACM, pp 7–18

  13. Orlin JB (1997) A polynomial time primal network simplex algorithm for minimum cost flows. Math Program 78(2):109–129

    Article  MathSciNet  MATH  Google Scholar 

  14. Goldfarb D, Hao JX (1990) A primal simplex algorithm that solves the maximum flow problem in at most nm pivots and O(n2m) time. Math Program 47(1–3):353–365

    Article  MATH  Google Scholar 

  15. Ciupala L (2006) A deficit scaling algorithm for the minimum flow problem. Sadhana 31(3):227–233

    Article  MathSciNet  Google Scholar 

  16. Holzhauser M, Krumke SO, Thielen C (2016) On the complexity and approximability of budget-constrained minimum cost flows. arXiv preprint arXiv:1607.02282

  17. Armstrong RD, Jin ZY (1997) A new strongly polynomial dual network simplex algorithm. Math Program 78(2):131–148

    Article  MathSciNet  MATH  Google Scholar 

  18. Eusebio A, Figueira JR, Ehrgott M (2009) A primal-dual simplex algorithm for bi-objective network flow problems. 4OR 7(3):255–273

    Article  MathSciNet  MATH  Google Scholar 

  19. Caliskan C (2008) A double scaling algorithm for the constrained maximum flow problem. Comput Oper Res 35(4):1138–1150

    Article  MathSciNet  MATH  Google Scholar 

  20. Holzhauser M, Krumke SO, Thielen C (2017) A network simplex method for the budget-constrained minimum cost flow problem. Eur J Oper Res 259(3):864–872

    Article  MathSciNet  MATH  Google Scholar 

  21. Çalışkan C (2011) A specialized network simplex algorithm for the constrained maximum flow problem. Eur J Oper Res 210(2):137–147

    Article  MathSciNet  MATH  Google Scholar 

  22. Hwu WM (2014) What is ahead for parallel computing. J Parallel Distrib Comput 74(7):2574–2581

    Article  Google Scholar 

  23. Sato M (2002) OpenMP: parallel programming API for shared memory multiprocessors and on-chip multiprocessors. In: Proceedings of the 15th International Symposium on System Synthesis. ACM, pp 109–111

  24. Kirk D (2007) NVIDIA CUDA software and GPU parallel computing architecture. In: ISMM, vol 7, pp 103–104

  25. Buck I (2007) GPU computing: programming a massively parallel processor. In: 2007. CGO’07. International Symposium on Code Generation and Optimization. IEEE, p 17

  26. Tziritas N, Khan SU, Xu CZ, Hong J (2012) An optimal fully distributed algorithm to minimize the resource consumption of cloud applications. In: 2012 IEEE 18th International Conference on Parallel and Distributed Systems (ICPADS). IEEE, pp 61–68

  27. Tziritas N, Khan SU, Xu CZ, Loukopoulos T, Lalis S (2013) On minimizing the resource consumption of cloud applications using process migrations. J Parallel Distrib Comput 73(12):1690–1704

    Article  Google Scholar 

  28. Bertsekas DP, Castanon DA (1993) Parallel primal–dual methods for the minimum cost flow problem. Comput Optim Appl 2(4):317–336

    Article  MathSciNet  MATH  Google Scholar 

  29. Beraldi P, Guerriero F (1997) A parallel asynchronous implementation of the ε-relaxation method for the linear minimum cost flow problem. Parallel Comput 23(8):1021–1044

    Article  MathSciNet  MATH  Google Scholar 

  30. Beraldi P, Guerriero F, Musmanno R (1997) Efficient parallel algorithms for the minimum cost flow problem. J Optim Theory Appl 95(3):501–530

    Article  MathSciNet  MATH  Google Scholar 

  31. Beraldi P, Guerriero F, Musmanno R (2001) Parallel algorithms for solving the convex minimum cost flow problem. Comput Optim Appl 18(2):175–190

    Article  MathSciNet  MATH  Google Scholar 

  32. Imai H, Iwano K (1990) Efficient sequential and parallel algorithms for planar minimum cost flow. In: Asano T, Ibaraki T, Imai H, Nishizeki T (eds) Algorithms. SIGAL 1990. Lecture Notes in Computer Science, vol 450. Springer, Berlin, Heidelberg, pp 21–30

    Google Scholar 

  33. Orlin JB, Stein C (1993) Parallel algorithms for the assignment and minimum-cost flow problems. Oper Res Lett 4(14):181–186

    Article  MathSciNet  MATH  Google Scholar 

  34. Andrzej L, Mia P (2012) A fast parallel algorithm for minimum-cost small integral flows. In: Euro-Par 2012 Parallel Processing. Springer, Berlin, pp 688–699

  35. Lingas A, Persson M (2015) A fast parallel algorithm for minimum-cost small integral flows. Algorithmica 72(2):607–619

    Article  MathSciNet  MATH  Google Scholar 

  36. Yarmish G, Slyke R (2009) A distributed, scaleable simplex method. J Supercomput 49(3):373–381

    Article  Google Scholar 

  37. Ploskas N, Samaras N, Margaritis K (2013) A parallel implementation of the revised simplex algorithm using OpenMP: some preliminary results. In: Optimization Theory, Decision Making, and Operational Research Applications, Springer Proceedings in Mathematics and Statistics, vol 31, pp 163–175

  38. Lalami ME, Boyer V, El-Baz D (2011) Efficient implementation of the simplex method on a CPU–GPU system. In: Proceedings of the 2011 IEEE International Symposium on Parallel and Distributed Processing Workshops and PhD Forum, Washington DC, 16–20 May 2011, pp 1999–2006

  39. Bieling J, Peschlow P, Martini P (2010) An efficient GPU implementation of the revised simplex method. In: Proceedings of the 24th IEEE International Parallel and Distributed Processing Symposium, 19–23 April 2010, Atlanta, pp 1–8

  40. Meyer X, Albuquerque P, Chopard B (2011) A MultiGPU implementation and performance model for the standard simplex method. In: Proceedings of the 1st International Symposium and 10th Balkan Conference on Operational Research, 22–25 September 2011, Thessaloniki, pp 312–319

  41. Tar P, Stágel B, Maros I (2017) Parallel search paths for the simplex algorithm. CEJOR 25(4):967–984

    Article  MathSciNet  MATH  Google Scholar 

  42. Hall JAJ, McKinnon KIM (1998) ASYNPLEX, an asynchronous parallel revised simplex algorithms. Ann Oper Res 81:27–50

    Article  MathSciNet  MATH  Google Scholar 

  43. Klabjan D, Johnson EL, Nemhauser GL (2000) A parallel primal–dual simplex algorithm. Oper Res Lett 27(2):47–55

    Article  MathSciNet  MATH  Google Scholar 

  44. Lubin M, Hall JAJ, Petra CG, Anitescu M (2013) Parallel distributed-memory simplex for large-scale stochastic LP problems. Comput Optim Appl 55(3):571–596

    Article  MathSciNet  MATH  Google Scholar 

  45. Thulasiraman K, Chalasani RP, Comeau MA (1993) Parallel network dual simplex method on a shared memory multiprocessor. In: Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing, 1993. IEEE, pp 408–415

  46. Mamalis B, Perlitis M (2016) A hybrid parallelization scheme for standard simplex method based on CPU/GPU collaboration. In: Proceedings of the 20th Pan-Hellenic Conference on Informatics, pp 1–6

  47. Huangfu Q, Hall JJ (2018) Parallelizing the dual revised simplex method. Math Program Comput 10(1):119–142

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhong Z, Feng M, Liu D (2015) Parallelization of revised simplex algorithm on GPUs. In: 2015 International Conference on Network and Information Systems for Computers. IEEE, pp 349–353

  49. He L, Bai H, Jiang Y, Ouyang D, Jiang S (2018) Revised simplex algorithm for linear programming on GPUs with CUDA. Multimed Tools Appl 77(22):30035–30050

    Article  Google Scholar 

  50. Goldberg AV, Tarjan RE (1988) A new approach to the maximum flow problem. J ACM 35(4):921–940

    Article  MathSciNet  MATH  Google Scholar 

  51. Harel D, Tarjan RE (1984) Fast algorithms for finding nearest common ancestors. SIAM J Comput 13(2):338–355

    Article  MathSciNet  MATH  Google Scholar 

  52. Eppstein D (1994) Clustering for faster network simplex pivots. In: SODA, pp 160–166

  53. Ahuja RK, Orlin JB (1996) Use of representative operation counts in computational testing of algorithms. Informs J Comput 8(3):318–330

    Article  MATH  Google Scholar 

  54. Ahuja RK, Orlin JB, Sharma P, Sokkalingam PT (2002) A network simplex algorithm with O(n) consecutive degenerate pivots. Oper Res Lett 30(3):141–148

    Article  MathSciNet  MATH  Google Scholar 

  55. Cormen TH et al (2001) Introduction to algorithms. MIT Press, Cambridge

    MATH  Google Scholar 

  56. Kelly DJ, ONeill GM (1991) The minimum cost flow problem and the network simplex solution method. Doctoral dissertation, University College Dublin, Graduate School of Business

  57. Badics T, Boros E, Cepek O (1993) Implementing a new maximum flow algorithm. In: Johnson DS, McGeoch CC (eds) Network flows and matching: first DIMACS implementation challenge, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol 12. American Mathematical Society, Providence

    Google Scholar 

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Acknowledgements

This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA19030301); the National Natural Science Foundation of China (No. 41701452 and 41974006); the Natural Science Foundation of Guangdong (No. 2019A1515011501); and the Shenzhen Scientific Research and Development Funding Program (No. KQJSCX20180328093453763, JCYJ20180305125101282).

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  1. Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, China

    Jincheng Jiang & Jinsong Chen

  2. Shenzhen Key Laboratory of Spatial Smart Sensing and Service, Research Institute for Smart Cities, School of Architecture and Urban Planning, Shenzhen University, Shenzhen, 518060, China

    Chisheng Wang

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Jiang, J., Chen, J. & Wang, C. Multi-granularity hybrid parallel network simplex algorithm for minimum-cost flow problems. J Supercomput 76, 9800–9826 (2020). https://doi.org/10.1007/s11227-020-03227-9

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