Sparse Approximation in Lattices and Semigroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914322535415808 |
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| author | Kuhlmann, Stefan Oertel, Timm Weismantel, Robert |
| author_facet | Kuhlmann, Stefan Oertel, Timm Weismantel, Robert |
| contents | This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number $k < n$, to approximate $b$ with $Ay$ where $y$ is an integer or non-negative integer solution with at most $k$ non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter $k$, then the paper explains why the quality of the approximation increases exponentially as $k$ goes to $n$. This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23990 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sparse Approximation in Lattices and Semigroups Kuhlmann, Stefan Oertel, Timm Weismantel, Robert Optimization and Control Discrete Mathematics Combinatorics This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number $k < n$, to approximate $b$ with $Ay$ where $y$ is an integer or non-negative integer solution with at most $k$ non-zero components. We establish upper bounds for this question in general. In specific cases, these bounds are tight. If we view the approximation quality as a function of the parameter $k$, then the paper explains why the quality of the approximation increases exponentially as $k$ goes to $n$. This paper is a complete version of an extended abstract that appeared at the 26th International Conference on Integer Programming and Combinatorial Optimization (IPCO). |
| title | Sparse Approximation in Lattices and Semigroups |
| topic | Optimization and Control Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2410.23990 |