Fault-Tolerant Matroid Bases

2025-06-27

AI Summary

This paper investigates the construction of $k$-fault-tolerant bases in matroids: finding a minimum-size set that remains a basis—i.e., retains full rank—after the removal of any $k$ elements. This problem unifies diverse fault-tolerant structures across vector spaces, graphs, and set systems. The authors present the first fixed-parameter tractable (FPT) algorithm for this problem, parameterized jointly by the matroid rank $r$ and the fault tolerance $k$. They fully characterize its computational complexity: polynomial-time solvable when $r leq 2$, Para-NP-hard when $r geq 3$, and already NP-hard for $k = 1$. This work establishes the first tight complexity landscape for the $k$-fault-tolerant matroid basis problem, resolving a fundamental open question in fault-tolerant matroid optimization and filling a key theoretical gap in the field.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Constructing fault-tolerant bases in matroids
• Finding minimum-size sets resilient to k removals
• Developing FPT algorithm for k and rank r

Innovation

• FPT algorithm for fault-tolerant bases
• Parameterized by k and matroid rank r
• Unifies fault-tolerant concepts in matroids