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Towards the Proximity Conjecture on Group-Labeled Matroids

Authors Dániel Garamvölgyi, Ryuhei Mizutani, Taihei Oki , Tamás Schwarcz , Yutaro Yamaguchi

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  • Mathematics of computing → Matroids and greedoids
Keywords
  • sparse paving matroid
  • subsequence-interchangeable base orderability
  • congruency constraint
  • multiple labelings

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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.

Cite As Get BibTex

Dániel Garamvölgyi, Ryuhei Mizutani, Taihei Oki, Tamás Schwarcz, and Yutaro Yamaguchi. Towards the Proximity Conjecture on Group-Labeled Matroids. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 85:1-85:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.85

Author Details

Dániel Garamvölgyi
  • MTA-ELTE Matroid Optimization Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Ryuhei Mizutani
  • Faculty of Science and Technology, Keio University, Yokohama, Japan
Taihei Oki
  • Institute for Chemical Reaction Design and Discovery (ICReDD), Hokkaido University, Sapporo, Japan
  • Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan
Tamás Schwarcz
  • Department of Mathematics, London School of Economics and Political Science, UK
Yutaro Yamaguchi
  • Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, Japan

Funding

This research has been implemented with the support provided by the Lendület Programme of the Hungarian Academy of Sciences - grant number LP2021-1/2021.
  • Mizutani, Ryuhei: Supported by JSPS KAKENHI Grant Number JP23KJ0379 and JST SPRING Grant Number JPMJSP2108.
  • Oki, Taihei: Supported by JST ERATO Grant Number JPMJER1903, JST CREST Grant Number JPMJCR24Q2, JST FOREST Grant Number JPMJFR232L, JSPS KAKENHI Grant Numbers JP22K17853 and 24K21315, and Start-up Research Funds in ICReDD, Hokkaido University.
  • Schwarcz, Tamás: Supported by EPSRC grant EP/X030989/1. Most of this work was done while this author was affiliated to the HUN-REN–ELTE Egerváry Research Group.
  • Yamaguchi, Yutaro: Supported by JSPS KAKENHI Grant Numbers 20K19743, 20H00605, and 25H01114, by JST CRONOS Japan Grant Number JPMJCS24K2, and by Start-up Program in Graduate School of Information Science and Technology, Osaka University.

Acknowledgements

The authors thank anonymous referees for careful reading. The authors are grateful to the organizers of the 16th Emléktábla Workshop, where the collaboration of the authors started. The authors thank Kristóf Bérczi, Siyue Liu, and Chao Xu for several helpful discussions.

Supplementary Materials

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