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Fuzzy Simultaneous Congruences

Authors Max A. Deppert , Klaus Jansen , Kim-Manuel Klein

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Max A. Deppert
  • Kiel University, Germany
Klaus Jansen
  • Kiel University, Germany
Kim-Manuel Klein
  • Kiel University, Germany

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Max A. Deppert, Klaus Jansen, and Kim-Manuel Klein. Fuzzy Simultaneous Congruences. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 39:1-39:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We introduce a very natural generalization of the well-known problem of simultaneous congruences. Instead of searching for a positive integer s that is specified by n fixed remainders modulo integer divisors a₁,… ,a_n we consider remainder intervals R₁,… ,R_n such that s is feasible if and only if s is congruent to r_i modulo a_i for some remainder r_i in interval R_i for all i. This problem is a special case of a 2-stage integer program with only two variables per constraint which is is closely related to directed Diophantine approximation as well as the mixing set problem. We give a hardness result showing that the problem is NP-hard in general. By investigating the case of harmonic divisors, i.e. a_{i+1}/a_i is an integer for all i < n, which was heavily studied for the mixing set problem as well, we also answer a recent algorithmic question from the field of real-time systems. We present an algorithm to decide the feasibility of an instance in time 𝒪(n²) and we show that if it exists even the smallest feasible solution can be computed in strongly polynomial time 𝒪(n³).

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Integer programming
  • Simultaneous congruences
  • Integer programming
  • Mixing Set
  • Real-time scheduling
  • Diophantine approximation


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  1. Manindra Agrawal and Somenath Biswas. Primality and identity testing via chinese remaindering. In Proc. FOCS 1999, pages 202-209, 1999. Google Scholar
  2. Saoussen Anssi, Stefan Kuntz, Sébastien Gérard, and François Terrier. On the gap between schedulability tests and an automotive task model. J. Syst. Archit., 59(6):341-350, 2013. Google Scholar
  3. Reuven Bar-Yehuda and Dror Rawitz. Efficient algorithms for integer programs with two variables per constraint. Algorithmica, 29(4):595-609, 2001. Google Scholar
  4. Sanjoy K. Baruah, Louis E. Rosier, and Rodney R. Howell. Algorithms and complexity concerning the preemptive scheduling of periodic, real-time tasks on one processor. Real-Time Systems, 2(4):301-324, 1990. Google Scholar
  5. Vincenzo Bonifaci, Alberto Marchetti-Spaccamela, Nicole Megow, and Andreas Wiese. Polynomial-time exact schedulability tests for harmonic real-time tasks. In Proc. RTSS 2013, pages 236-245, 2013. Google Scholar
  6. Michele Conforti, Gerard Cornuéjols, and Giacomo Zambelli. Integer Programming. Springer, 2014. Google Scholar
  7. Michele Conforti, Marco Di Summa, and Laurence A. Wolsey. The mixing set with flows. SIAM J. Discrete Math., 21(2):396-407, 2007. Google Scholar
  8. Michele Conforti, Marco Di Summa, and Laurence A. Wolsey. The mixing set with divisible capacities. In Proc. IPCO 2008, pages 435-449, 2008. Google Scholar
  9. Michele Conforti and Giacomo Zambelli. The mixing set with divisible capacities: A simple approach. Oper. Res. Lett., 37(6):379-383, 2009. Google Scholar
  10. Jana Cslovjecsek, Friedrich Eisenbrand, Christoph Hunkenschröder, Lars Rohwedder, and Robert Weismantel. Block-structured integer and linear programming in strongly polynomial and near linear time, 2020 (Manuscript). URL:
  11. Friedrich Eisenbrand, Karthikeyan Kesavan, Raju S. Mattikalli, Martin Niemeier, Arnold W. Nordsieck, Martin Skutella, José Verschae, and Andreas Wiese. Solving an avionics real-time scheduling problem by advanced ip-methods. In Mark de Berg and Ulrich Meyer, editors, Proc. ESA 2010, volume 6346 of Lecture Notes in Computer Science, pages 11-22, 2010. Google Scholar
  12. Friedrich Eisenbrand and Günter Rote. Fast 2-variable integer programming. In Proc. IPCO 2001, pages 78-89, 2001. Google Scholar
  13. Friedrich Eisenbrand and Thomas Rothvoß. New hardness results for diophantine approximation. In Proc. APPROX 2009, pages 98-110, 2009. Google Scholar
  14. Oded Goldreich, Dana Ron, and Madhu Sudan. Chinese remaindering with errors. In Proc. STOC 1999, pages 225-234, 1999. Google Scholar
  15. Oktay Günlük and Yves Pochet. Mixing mixed-integer inequalities. Math. Program., 90(3):429-457, 2001. Google Scholar
  16. Venkatesan Guruswami, Amit Sahai, and Madhu Sudan. "soft-decision" decoding of chinese remainder codes. In Proc. FOCS 2000, pages 159-168, 2000. Google Scholar
  17. Raymond Hemmecke, Shmuel Onn, and Lyubov Romanchuk. n-fold integer programming in cubic time. Math. Program., 137(1-2):325-341, 2013. Google Scholar
  18. Raymond Hemmecke and Rüdiger Schultz. Decomposition of test sets in stochastic integer programming. Math. Program., 94(2-3):323-341, 2003. Google Scholar
  19. Klaus Jansen, Alexandra Lassota, and Lars Rohwedder. Near-linear time algorithm for n-fold ilps via color coding. In Proc. ICALP 2019, pages 75:1-75:13, 2019. Google Scholar
  20. Kim-Manuel Klein. About the complexity of two-stage stochastic ips. In Daniel Bienstock and Giacomo Zambelli, editors, Proc. IPCO 2020, volume 12125 of Lecture Notes in Computer Science, pages 252-265, 2020. Google Scholar
  21. Donald E. Knuth. The Art of Computer Programming, Volume II: Seminumerical Algorithms, 2nd Edition. Prentice Hall, 1981. Google Scholar
  22. J. C. Lagarias. The computational complexity of simultaneous diophantine approximation problems. SIAM J. Comput., 14(1):196-209, 1985. Google Scholar
  23. Joseph Y.-T. Leung and Jennifer Whitehead. On the complexity of fixed-priority scheduling of periodic, real-time tasks. Perform. Evaluation, 2(4):237-250, 1982. Google Scholar
  24. Andrew J. Miller and Laurence A. Wolsey. Tight formulations for some simple mixed integer programs and convex objective integer programs. Math. Program., 98(1-3):73-88, 2003. Google Scholar
  25. Thi Huyen Chau Nguyen, Werner Grass, and Klaus Jansen. Exact polynomial time algorithm for the response time analysis of harmonic tasks with constrained release jitter, 2019 (Manuscript). URL:
  26. Yves Pochet and Laurence A. Wolsey. Production Planning by Mixed Integer Programming (Springer Series in Operations Research and Financial Engineering). Springer, Berlin, Heidelberg, 2006. Google Scholar
  27. Chi-Sheng Shih, Sathish Gopalakrishnan, Phanindra Ganti, Marco Caccamo, and Lui Sha. Template-based real-time dwell scheduling with energy constraint. In Proc. RTAS 2003, page 19, 2003. Google Scholar
  28. Mikael Sjödin and Hans Hansson. Improved response-time analysis calculations. In Proc. RTSS 1998, pages 399-408, 1998. Google Scholar
  29. Yang Xu, Anton Cervin, and Karl-Erik Årzén. Lqg-based scheduling and control co-design using harmonic task periods. Technical report, Department of Automatic Control, Lund Institute of Technology, Lund University, August 2016. URL:
  30. Ming Zhao and Ismael R. de Farias Jr. The mixing-mir set with divisible capacities. Math. Program., 115(1):73-103, 2008. Google Scholar
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