Fuzzy同时同余
Fuzzy Simultaneous Congruences
摘要：
我们介绍了一个非常自然的推广著名的问题，同时同余。代替搜索由$n$固定的余数模整数除数$a_1,\dots,a_n$指定的正整数$s$，我们考虑余数区间$R_1,\dots,R_n$，使得$s$是可行的当且仅当对于所有$i$，$s$对于区间$R_i$中的某个余数$r_i$全等于$r_i$模$a_i$。这个问题是一个特殊的情况下，只有两个变量的2阶段整数规划的约束，这是密切相关的有向丢番图逼近以及混合集问题。我们给出了一个硬度的结果表明，该问题是NP-困难的一般。通过研究调和因子的情况，即$a_{i+1}/a_i$是一个整数的所有$i<n$，这是大量研究的混合集问题，以及，我们也回答了最近的算法问题，从实时系统领域。我们提出了一个算法来决定一个实例的时间$\mathcal{O}(n^2)$的可行性，我们表明，如果它存在，即使是最小的可行解可以计算在强多项式时间$\mathcal{O}(n^3)$。
英文摘要：
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. 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 $\mathcal{O}(n^2)$ and we show that if it exists even the smallest feasible solution can be computed in strongly polynomial time $\mathcal{O}(n^3)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
