We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor gamma(n) > 1 in random exponential time 2^{O(n)}, it is in AM for gamma(n) = 2, in coAM for gamma(n) = sqrt{n / log n}, and in NP intersected coNP for gamma(n) = sqrt{n}. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor gamma(n) = log n, but cannot be solved in polynomial time for some gamma(n) = Omega(log log n) unless NP can be simulated in deterministic n^{O(log log log n)} time. Moreover, we prove that the problem is NP-hard for any constant approximation factor, it is Pi_2-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be Pi_2-hard for any constant approximation factor. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a AM protocol achieving approximation factor gamma(n) = sqrt{n / log n}, solving an open problem of Blomer and Seifert (STOC'99).