Fault-Tolerant Matroid Bases

TL;DR

This paper introduces a fixed-parameter tractable algorithm for constructing fault-tolerant bases in matroids, extending fault-tolerance concepts across various structures and analyzing computational complexity based on parameters k and r.

Contribution

It presents the first FPT algorithm for the k-fault-tolerant basis problem parameterized by k and r, and establishes complexity bounds relative to these parameters.

Findings

The problem is NP-hard for k=1.

It is Para-NP-hard for r ≥ 3.

It is polynomial-time solvable for r ≤ 2.

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 \geq 3 and polynomial-time solvable for r \leq 2.