Fuzzy Simultaneous Congruences
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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,...,a_n we consider remainder intervals R_1,...,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<n. We present an algorithm to decide the feasibility of an instance in time O(n^2) and we show that even the smallest feasible solution can be computed in strongly polynomial time O(n^3).
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