TITLE: Finding covering system of congruences with prescribed moduli
QUESTION [5 upvotes]: Is there an algorithm (better than exhaustive search) that for a given list of moduli $m_i$ produces a list of residues $0\leq r_i<m_i$ such that $(m_i,r_i)$ is a covering system of congruences, or determines that none exists? The covering system is in the sense of Erdős: every integer satisfies $n=r_i$ (mod $m_i$) for at least one $i$. The moduli on the list may repeat, but can only be used as many times as the list prescribes. 
Also, are there good necessary and/or sufficient conditions for existence of a cover, given the list of moduli, better than that the sum of the moduli reciprocals has to exceed $1$? For example, I am pretty sure there is no cover for $2,3,5,5$ that satisfies it, and I couldn't find a cover even for $2,3,5,5,6,10,30$

REPLY [5 votes]: There is some discussion of this sort of question in Jenkin and Simpson, Composite Covering Systems of Minimum Cardinality, available here. In particular, it says that given a collection $$\{\,(m_1,a_1),\dots,(m_n,a_n)\,\}$$ of ordered pairs of positive integers, the question of whether there is an integer $x$ satisfying none of the congruences $x\equiv a_i\pmod{m_i}$ is NP-complete. References are given to the Garey & Johnson book, and to Stockmeyer, L.J., and Meyer, A.R., Word problems requiring exponential time, Proc. 5th Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York(1973), 1-9.