The complexity of the covering radius problem

Venkatesan Guruswami, Daniele Micciancio, Oded Regev

Research output: Contribution to journalArticle

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 γ (n) = √n/log n, and in NP ∩ coNP 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 (loglog n)unless NP can be simulated in deterministic nO(logloglogn) time. Moreover, we prove that the problem is NP-hard for any constant approximation factor, it is Π2-hard for some constant approximation factor, and that it is unlikely to be Π2-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 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 nondeterministic polynomial time reduction to the closest vector problem.

Original languageEnglish (US)
Pages (from-to)90-121
Number of pages32
JournalComputational Complexity
Volume14
Issue number2
DOIs
StatePublished - Jun 2005

Fingerprint

Covering Radius
Polynomials
Computational complexity
Network protocols
Approximation
Polynomial time
Hardness
Linear Codes
n-dimensional
Computational Complexity
Hardness of Approximation
Polynomial Hierarchy
Exponential time
Open Problems

Keywords

  • Approxiamation algorithms
  • Complexity classes
  • Covering radius
  • Hardness of approximation
  • Interactive proofs
  • Lattices
  • Linear codes
  • Polynomial time hierachy

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Mathematics(all)
  • Computational Mathematics
  • Theoretical Computer Science

Cite this

The complexity of the covering radius problem. / Guruswami, Venkatesan; Micciancio, Daniele; Regev, Oded.

In: Computational Complexity, Vol. 14, No. 2, 06.2005, p. 90-121.

Research output: Contribution to journalArticle

Guruswami, Venkatesan ; Micciancio, Daniele ; Regev, Oded. / The complexity of the covering radius problem. In: Computational Complexity. 2005 ; Vol. 14, No. 2. pp. 90-121.
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