On circuit diameter bounds via circuit imbalances

Koh, Zhuan Khye, Natura, Bento and Végh, László A. ORCID: 0000-0003-1152-200X (2024) On circuit diameter bounds via circuit imbalances. Mathematical Programming, 206 (1-2). 631 - 662. ISSN 0025-5610

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Abstract

We study the circuit diameter of polyhedra, introduced by Borgwardt, Finhold, and Hemmecke (SIAM J. Discrete Math. 29(1), 113–121 (2015)) as a relaxation of the combinatorial diameter. We show that the circuit diameter of a system {x∈R n:Ax=b,0≤x≤u} for A∈R m×n is bounded by O(mmin{m,n-m}log(m+κ A)+nlogn), 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 2log(n+κ A)) augmentation steps.

Item Type:	Article
Official URL:	https://link.springer.com/journal/10107
Additional Information:	© 2024 The Author(s)
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	19 Jun 2024 09:18
Last Modified:	16 Nov 2024 03:51
URI:	http://eprints.lse.ac.uk/id/eprint/123916

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