Towards the proximity conjecture on group-labeled matroids

Garamvölgyi, Dániel, Mizutani, Ryuhei, Oki, Taihei, Schwarcz, Tamás and Yamaguchi, Yutaro (2025) Towards the proximity conjecture on group-labeled matroids. In: Censor-Hillel, Keren, Grandoni, Fabrizio, Ouaknine, Joel and Puppis, Gabriele, (eds.) 52nd International Colloquium on Automata, Languages, and Programming, ICALP 2025. Leibniz International Proceedings in Informatics, LIPIcs. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. ISBN 9783959773720

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Abstract

Consider a matroid M whose ground set is equipped with a labeling to an abelian group. A basis of M is called F-avoiding if the sum of the labels of its elements is not in a forbidden label set F. Hörsch, Imolay, Mizutani, Oki, and Schwarcz (2024) conjectured that if an F-avoiding basis exists, then any basis can be transformed into an F-avoiding basis by exchanging at most |F| elements. This proximity conjecture is known to hold for certain specific groups; in the case where |F| ≤ 2; or when the matroid is subsequence-interchangeably base orderable (SIBO), which is a weakening of the so-called strongly base orderable (SBO) property. In this paper, we settle the proximity conjecture for sparse paving matroids or in the case where |F| ≤ 4. Related to the latter result, we present the first known example of a non-SIBO matroid. We further address the setting of multiple group-label constraints, showing proximity results for the cases of two labelings, SIBO matroids, matroids representable over a fixed, finite field, and sparse paving matroids.

Item Type:	Book Section
Additional Information:	© The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	15 Jul 2025 13:57
Last Modified:	16 Jul 2025 23:19
URI:	http://eprints.lse.ac.uk/id/eprint/128838

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