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Sparse representation of vectors in lattices and semigroups
Mathematical Programming ( IF 2.7 ) Pub Date : 2021-05-04 , DOI: 10.1007/s10107-021-01657-8
Iskander Aliev , Gennadiy Averkov , Jesús A. De Loera , Timm Oertel

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 \(\ell _0\)-norm of the vector. Our main results are new improved bounds on the minimal \(\ell _0\)-norm of solutions to systems \(A\varvec{x}=\varvec{b}\), where \(A\in \mathbb {Z}^{m\times n}\), \({\varvec{b}}\in \mathbb {Z}^m\) and \(\varvec{x}\) 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 \(\ell _0\)-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 \(\mathbb {R}\), to other subdomains such as \(\mathbb {Z}\). 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.



中文翻译:

向量在格和半群中的稀疏表示

我们研究具有和不具有非负性约束的线性Diophantine方程组解的稀疏性。解向量的稀疏度是其非零项的数量,称为向量的\(\ ell _0 \)-范数。我们的主要结果是对系统\(A \ varvec {x} = \ varvec {b} \)的最小\(\ ell _0 \)-范数的新改进边界,其中\(A \ in \ mathbb {Z } ^ {m \ times n} \)\({\ varvec {b}} \\\ mathbb {Z} ^ m \)\(\ varvec {x} \)都是一般的整数向量(晶格情况) )或非负整数向量(半群情况)。在某些情况下,我们使用多项式时间算法来计算具有\(\ ell _0 \)的解决方案-范数满足获得的界限。我们证明我们的界限是紧密的。我们的边界可以看作是自然地将\(\ mathbb {R} \)上矩阵的秩推广到\(\ mathbb {Z} \)等其他子域的函数。我们表明,这些新的类似秩的函数通常都难于计算NP,但是对于固定数量的变量而言,多项式时间是可计算的。

更新日期:2021-05-04
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