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).