Is it know whether finding an exact covering system, if it exists, given a set of moduli is NP-Hard? That is, given $n_1,...,n_k \in \mathbb{Z}$, find $a_1,...,a_k \in \mathbb{Z}$ such that the congruence $a_i \mod n_i$ partition the integers.
Related to this question: Finding covering system of congruences with prescribed moduli but I am looking for exact systems, i.e. it may be assumed that $\sum_i \frac1{n_i} =1$. This should be easier since we can check whether a set of $a_i$'s gives an exact cover with Bezout's identity.
In this book it is stated that characterizing exact cover is an outstanding problem. However, here a recursive definition of exact covers in terms of smaller exact covers is given. The construction is easy to check though probably hard to find, but it seems to me that this gives a characterization. Would a good characterization require that we can find it easily too?
