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On Circuit Diameter Bounds via Circuit Imbalances

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Integer Programming and Combinatorial Optimization (IPCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13265))

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Abstract

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 \(\{x\in \mathbb {R}^n:\, Ax=b,\, \mathbb {0}\le x\le u\}\) for \(A\in \mathbb {R}^{m\times n}\) is bounded by \(O(m^2\log (m+\kappa _A)+n\log n)\), where \(\kappa _A\) 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 \(O(n^3\log (n+\kappa _A))\) augmentation steps.

This is an extended abstract. The full version of the paper with all proofs is available on arXiv:2111.07913. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements ScaleOpt–757481 and QIP–805241). This work was done while the authors participated in the Discrete Optimization Trimester Program at the Hausdorff Institute for Mathematics in Bonn in 2021.

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Authors and Affiliations

  1. Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

    Daniel Dadush

  2. London School of Economics and Political Science, London, UK

    Zhuan Khye Koh, Bento Natura & László A. Végh

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  1. Daniel Dadush
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  2. Zhuan Khye Koh
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  3. Bento Natura
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  4. László A. Végh
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Correspondence to Zhuan Khye Koh .

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Editors and Affiliations

  1. Delft University of Technology, Delft, The Netherlands

    Karen Aardal

  2. Eindhoven University of Technology, Eindhoven, The Netherlands

    Laura Sanità

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Dadush, D., Koh, Z.K., Natura, B., Végh, L. (2022). On Circuit Diameter Bounds via Circuit Imbalances. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_11

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  • DOI: https://doi.org/10.1007/978-3-031-06901-7_11

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