On Matrices over a Polynomial Ring with Restricted Subdeterminants
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914706746245120 |
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| author | Celaya, Marcel Kuhlmann, Stefan Weismantel, Robert |
| author_facet | Celaya, Marcel Kuhlmann, Stefan Weismantel, Robert |
| contents | This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We investigate in particular matrices whose subdeterminants all lie in a fixed set $S\subseteq\mathbb{Z}[x]$. Such matrices, which we call totally $S$-modular matrices, are closed with respect to taking submatrices, so it is natural to look at minimally non-totally $S$-modular matrices which we call forbidden minors for $S$. Among other results, we prove that if $S$ is finite, then the set of all determinants attained by a forbidden minor for $S$ is also finite. Specializing to the integers, we subsequently obtain the following positive complexity result: the recognition problem for totally $\pm\{0,1,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{-3,-2,1,2\}$ and the integer linear optimization problem for totally $\pm\{ 0,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{ -2,1\}$ can be solved in polynomial time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03845 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Matrices over a Polynomial Ring with Restricted Subdeterminants Celaya, Marcel Kuhlmann, Stefan Weismantel, Robert Optimization and Control Data Structures and Algorithms Combinatorics This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We investigate in particular matrices whose subdeterminants all lie in a fixed set $S\subseteq\mathbb{Z}[x]$. Such matrices, which we call totally $S$-modular matrices, are closed with respect to taking submatrices, so it is natural to look at minimally non-totally $S$-modular matrices which we call forbidden minors for $S$. Among other results, we prove that if $S$ is finite, then the set of all determinants attained by a forbidden minor for $S$ is also finite. Specializing to the integers, we subsequently obtain the following positive complexity result: the recognition problem for totally $\pm\{0,1,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{-3,-2,1,2\}$ and the integer linear optimization problem for totally $\pm\{ 0,a,a+1,2a+1\}$-modular matrices with $a\in\mathbb{Z}\backslash\{ -2,1\}$ can be solved in polynomial time. |
| title | On Matrices over a Polynomial Ring with Restricted Subdeterminants |
| topic | Optimization and Control Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2311.03845 |