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Fault-Tolerant Matroid Bases

Authors Matthias Bentert , Fedor V. Fomin , Petr A. Golovach , Laure Morelle

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ACM Subject Classification
  • Mathematics of computing → Combinatorial algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • Parameterized Complexity
  • matroids
  • robust bases

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Abstract

We investigate the problem of constructing fault-tolerant bases in matroids. Given a matroid ℳ and a redundancy parameter k, a k-fault-tolerant basis is a minimum-size set of elements such that, even after the removal of any k elements, the remaining subset still spans the entire ground set. Since matroids generalize linear independence across structures such as vector spaces, graphs, and set systems, this problem unifies and extends several fault-tolerant concepts appearing in prior research.
Our main contribution is a fixed-parameter tractable (FPT) algorithm for the k-fault-tolerant basis problem, parameterized by both k and the rank r of the matroid. This two-variable parameterization by k + r is shown to be tight in the following sense. On the one hand, the problem is already NP-hard for k = 1. On the other hand, it is Para-NP-hard for r ≥ 3 and polynomial-time solvable for r ≤ 2.

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Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, and Laure Morelle. Fault-Tolerant Matroid Bases. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.83

Author Details

Matthias Bentert
  • University of Bergen, Norway
  • Technische Universität Berlin, Germany
Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Laure Morelle
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France

Funding

The research leading to these results have been supported by the Research Council of Norway via the Franco-Norwegian AURORA project (grant no. 349476).
  • Bentert, Matthias: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416) and ERC Consolidator grant AdjustNet (agreement No. 864228).
  • Fomin, Fedor V.: Supported by the Research Council of Norway under BWCA project (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway under BWCA project (grant no. 314528).

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