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 $\max\{a\_i,b\_i~|~i\in[k]\}$.



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


