On Circuit Diameter Bounds via Circuit Imbalances
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| author | Dadush, Daniel Koh, Zhuan Khye Natura, Bento Végh, László A. |
| author_facet | Dadush, Daniel Koh, Zhuan Khye Natura, Bento Végh, László A. |
| contents | We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$ for $A \in \mathbb{R}^{m \times n}$ is bounded by $O(m \min\{m, n-m\} \log(m+ κ_A)+n \log n)$, where $κ_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of $A$ have polynomially bounded encoding length in $n$. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $O(mn^2\log(n+κ_A))$ augmentation steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_07913 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Circuit Diameter Bounds via Circuit Imbalances Dadush, Daniel Koh, Zhuan Khye Natura, Bento Végh, László A. Optimization and Control Discrete Mathematics Combinatorics We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIDMA 2015) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system $\{x \in \mathbb{R}^n: Ax=b, 0\leq x\leq u\}$ for $A \in \mathbb{R}^{m \times n}$ is bounded by $O(m \min\{m, n-m\} \log(m+ κ_A)+n \log n)$, where $κ_A$ is the circuit imbalance measure of the constraint matrix. This yields a strongly polynomial circuit diameter bound if e.g., all entries of $A$ have polynomially bounded encoding length in $n$. Further, we present circuit augmentation algorithms for LPs using the minimum-ratio circuit cancelling rule. Even though the standard minimum-ratio circuit cancelling algorithm is not finite in general, our variant can solve an LP in $O(mn^2\log(n+κ_A))$ augmentation steps. |
| title | On Circuit Diameter Bounds via Circuit Imbalances |
| topic | Optimization and Control Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2111.07913 |