License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.39
URN: urn:nbn:de:0030-drops-144792
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14479/
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Deppert, Max A. ; Jansen, Klaus ; Klein, Kim-Manuel

Fuzzy Simultaneous Congruences

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LIPIcs-MFCS-2021-39.pdf (0.8 MB)

Abstract

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 ?(n²) and we show that if it exists even the smallest feasible solution can be computed in strongly polynomial time ?(n³).

BibTeX - Entry

@InProceedings{deppert_et_al:LIPIcs.MFCS.2021.39,
  author =	{Deppert, Max A. and Jansen, Klaus and Klein, Kim-Manuel},
  title =	{{Fuzzy Simultaneous Congruences}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{39:1--39:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14479},
  URN =		{urn:nbn:de:0030-drops-144792},
  doi =		{10.4230/LIPIcs.MFCS.2021.39},
  annote =	{Keywords: Simultaneous congruences, Integer programming, Mixing Set, Real-time scheduling, Diophantine approximation}
}

Keywords: Simultaneous congruences, Integer programming, Mixing Set, Real-time scheduling, Diophantine approximation
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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