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 $\max\{a_i,b_i~|~i\in[k]\}$.