Fuzzy Simultaneous Congruences

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, . . . , an we consider remainder intervals R1, . . . , Rn such that s is feasible if and only if s is congruent to ri modulo ai for some remainder ri in interval Ri 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 realtime systems we investigate the case of harmonic divisors, i.e. ai+1/ai 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 ).