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Title: Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the$$\ell _0$$0-norm of the vector. Our main results are new improved bounds on the minimal$$\ell _0$$0-norm of solutions to systems$$A\varvec{x}=\varvec{b}$$Ax=b, where$$A\in \mathbb {Z}^{m\times n}$$AZm×n,$${\varvec{b}}\in \mathbb {Z}^m$$bZmand$$\varvec{x}$$xis either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with$$\ell _0$$0-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over$$\mathbb {R}$$R, to other subdomains such as$$\mathbb {Z}$$Z. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

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Publication Date:
Journal Name:
Mathematical Programming
Page Range or eLocation-ID:
p. 519-546
Springer Science + Business Media
Sponsoring Org:
National Science Foundation
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