The complexity of the covering radius problem on lattices and codes

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 γ(n) > 1 in random exponential time 2 ^O(n), it is in AM for γ(n) = 2, in coAM for γ(n) = √n/ log n, and in NP ∩ coMP for γ(n) = √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 γ(n) = log n, but cannot be solved in polynomial time for some γ(n) = Ω(log log n) unless MP can be simulated in deterministic n ^{O(log log log n)} time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is II ₂-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be II ₂-hard for approximation factors larger than 2. 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 γ(n) = √n/ log n, solving an open problem of Blömer and Seifert (STOC'99), and prove that the problem is also in coNP for γ(n) = √n. Both results are obtained by giving a gap-preserving non-deterministic polynomial time reduction to the closest vector problem.