# Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$

Let $$A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$$, where $$a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$$ and $$b_1, \ldots, b_k \in \mathbb{N}$$ be a system of arithmetic sequences.

For a positive integer $$m$$, system $$A$$ is called a $$m$$-cover of $$\mathbb{N}$$, if every natural number is covered by $$A$$ at least $$m$$ times.

My question is as follows: is there an efficient algorithm that given $$A$$ and $$m$$ decides if $$A$$ is a $$m$$-cover?

By an efficient algorithm I would perhaps mean an algorithm with the running time polynomial in $$k$$, $$m$$, and $$\max\{a_i,b_i~|~i\in[k]\}$$.

In 1973， L. J. Stockmeyer and A. R. Meyer [Proc. 5th. Ann. ACM Symp. on Theory of Computing, Assoc. for Computing Machinery] proved that the question whether a given system $$A=\{a_i+b_i\mathbb N\}_{i=1}^k$$ is a cover of $$\mathbb Z$$ (i.e., $$1$$-cover) is co-NP-complete. Thus NP=P if and only if we can decide whether $$A=\{a_i+b_i\mathbb N\}_{i=1}^k$$ is a cover of $$\mathbb Z$$ in polynomial time. Whether NP=P or not is a famous open problem.