New NP-Complete Problems Associated with Lattices

Shunichi HAYASHI  Mitsuru TADA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E90-A    No.5    pp.941-948
Publication Date: 2007/05/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e90-a.5.941
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
NP-complete problem,  lattice,  ELVP,  BELVP,  NELVP,  

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In this paper, we introduce a new decision problem associated with lattices, named the Exact Length Vector Problem (ELVP), and prove the NP-completeness of ELVP in the norm. Moreover, we define two variants of ELVP. The one is a binary variant of ELVP, named the Binary Exact Length Vector Problem (BELVP), and is shown to be NP-complete in any p norm (1 ≤ p ≤ ∞). The other is a nonnegative variant of ELVP, named the Nonnegative Exact Length Vector Problem (NELVP). NELVP is defined in the 1 norm, and is also shown to be NP-complete.