Sparse representation of vectors in lattices and semigroups

• Published in: Mathematical programming Vol. 192; no. 1-2; pp. 519 - 546
• Main Authors: Aliev, Iskander, Averkov, Gennadiy, De Loera, Jesús A., Oertel, Timm
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022; Springer; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Full Length Paper; Integers; Lattices (mathematics); Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Polynomials; Semigroups; Sparsity; Theoretical; 90C10; 20M10; 94A12; 52C07
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-021-01657-8

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 ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal ℓ 0 -norm of solutions to systems A x = b , where A ∈ Z m × n , b ∈ Z m and x is 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 ℓ 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 R , to other subdomains such as 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.