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A Federated Byzantine Agreement System is defined as a pair $(V, Q)$comprising a set of nodes $V$ and a quorum function $Q: V \mapsto 2^{2^{V}}\setminus \{\emptyset\}$ specifying for each node a set of subsets of nodes,called quorum slices. A subset of nodes is a quorum if and only if for each ofits nodes it also contains at least one of its quorum slices. The DisjointQuorums Problem answers the question whether a given instance of \acrlong{fbas}contains two quorums that have no nodes in common. We show that this problem is$\mathsf{NP-complete}$. We also study the problem of finding a quorum ofminimal size and show it is $\mathsf{NP-hard}$. Further, we consider theproblem of checking whether a given subset of nodes contains a quorum for someselected node. We show this problem is $\mathsf{P-complete}$ and describe amethod that solves it in linear time with respect to number of nodes and thetotal size of all quorum slices. Moreover, we analyze the complexity of some ofthese problems using the parametrized point of view.

Complexity of the quorum intersection property of the Federated Byzantine Agreement System