The k-fault-tolerant basis problem for matroids is FPT with parameter k plus rank r, NP-hard for k=1, and NP-hard for every fixed rank at least 3.
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Fault-Tolerant Matroid Bases
The k-fault-tolerant basis problem for matroids is FPT with parameter k plus rank r, NP-hard for k=1, and NP-hard for every fixed rank at least 3.