Space-Efficient Algorithm for Integer Programming with Few Constraints
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909306792706048 |
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| author | Rohwedder, Lars Węgrzycki, Karol |
| author_facet | Rohwedder, Lars Węgrzycki, Karol |
| contents | Integer linear programs $\min\{c^T x : A x = b, x \in \mathbb{Z}^n_{\ge 0}\}$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$, and $c \in \mathbb{Z}^n$, can be solved in pseudopolynomial time for any fixed number of constraints $m = O(1)$. More precisely, in time $(mΔ)^{O(m)} \text{poly}(I)$, where $Δ$ is the maximum absolute value of an entry in $A$ and $I$ the input size.
Known algorithms rely heavily on dynamic programming, which leads to a space complexity of similar order of magnitude as the running time. In this paper, we present a polynomial space algorithm that solves integer linear programs in $(mΔ)^{O(m (\log m + \log\logΔ))} \text{poly}(I)$ time, that is, in almost the same time as previous dynamic programming algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Space-Efficient Algorithm for Integer Programming with Few Constraints Rohwedder, Lars Węgrzycki, Karol Data Structures and Algorithms Integer linear programs $\min\{c^T x : A x = b, x \in \mathbb{Z}^n_{\ge 0}\}$, where $A \in \mathbb{Z}^{m \times n}$, $b \in \mathbb{Z}^m$, and $c \in \mathbb{Z}^n$, can be solved in pseudopolynomial time for any fixed number of constraints $m = O(1)$. More precisely, in time $(mΔ)^{O(m)} \text{poly}(I)$, where $Δ$ is the maximum absolute value of an entry in $A$ and $I$ the input size. Known algorithms rely heavily on dynamic programming, which leads to a space complexity of similar order of magnitude as the running time. In this paper, we present a polynomial space algorithm that solves integer linear programs in $(mΔ)^{O(m (\log m + \log\logΔ))} \text{poly}(I)$ time, that is, in almost the same time as previous dynamic programming algorithms. |
| title | Space-Efficient Algorithm for Integer Programming with Few Constraints |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2409.03681 |