The complexity of the covering radius problem
 Venkatesan Guruswami,
 Daniele Micciancio,
 Oded Regev
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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 ndimensional 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 ndimensional 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 NPhard 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 gappreserving nondeterministic polynomial time reduction to the closest vector problem.
 Title
 The complexity of the covering radius problem
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

computational complexity
Volume 14, Issue 2 , pp 90121
 Cover Date
 20050601
 DOI
 10.1007/s000370050193y
 Print ISSN
 10163328
 Online ISSN
 14208954
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Keywords

 Lattices
 linear codes
 covering radius
 approximation algorithms
 hardness of approximation
 complexity classes
 polynomial time hierarchy
 interactive proofs
 68Q17
 68Q25
 11H06
 11H31
 94B05
 Industry Sectors
 Authors

 Venkatesan Guruswami ^{(1)}
 Daniele Micciancio ^{(2)}
 Oded Regev ^{(3)}
 Author Affiliations

 1. Department of Computer Science, University of Washington, Seattle, WA, 98195, U.S.A
 2. Department of Computer Science and Engineering, University of California, San Diego La Jolla, CA, 92093, U.S.A
 3. Department of Computer Science, TelAviv University, TelAviv, 69978, Israel