# The complexity of the covering radius problem

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## Abstract.

We initiate the study of the computational complexity of the covering radius problem for 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 γ(*n*) > 1 in random exponential time 2^{O(n)}. We also prove that suitably defined gap versions of the problem lie in AM for λ(*n*) = 2, in coAM for
\(
\gamma (n) = {\sqrt {n/\log n} },
\) and in NP ∩ 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 Π_{2}-hard for *some* constant approximation factor, and that it is unlikely to be Π_{2}-hard for approximation factors larger than 2 (by giving an AM protocol for the appropriate gap problem). This is a natural hardness of approximation result in the polynomial hierarchy.

For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor
\(
\gamma (n) = {\sqrt {n/\log n} },
\) solving an open problem of Blömer and Seifert (STOC’99), and prove that the problem is also in coNP for
\(
\gamma (n) = {\sqrt n }.
\) Both results are obtained by giving a gap-preserving *nondeterministic* polynomial time reduction to the closest vector problem.