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Fuzzy Simultaneous Congruences
arXiv - CS - Discrete Mathematics Pub Date : 2020-02-18 , DOI: arxiv-2002.07746
Max A. Deppert; Klaus Jansen; Kim-Manuel Klein

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 $a_1,\dots,a_n$ we consider remainder intervals $R_1,\dots,R_n$ 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. Motivated by the study of the mixing set problem and a recent result in the field of real-time systems we investigate the case of harmonic divisors, i.e. $a_{i+1}/a_i$ is an integer for all $i
更新日期:2020-02-19

 

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