Papers › Complexity of the quorum intersection property of the Federated Byzantine Agreement System
Complexity of the quorum intersection property of the Federated Byzantine Agreement System
Łukasz Lachowski
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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 ↦2^(2ⱽ) ∖{∅} 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 of its nodes it also contains at least one of its quorum slices. The Disjoint Quorums 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 𝖭𝖯-𝖼𝗈𝗆𝗉𝗅𝖾𝗍𝖾. We also study the problem of finding a quorum of minimal size and show it is 𝖭𝖯-𝗁𝖺𝗋𝖽. Further, we consider the problem of checking whether a given subset of nodes contains a quorum for some selected node. We show this problem is 𝖯-𝖼𝗈𝗆𝗉𝗅𝖾𝗍𝖾 and describe a method that solves it in linear time with respect to number of nodes and the total size of all quorum slices. Moreover, we analyze the complexity of some of these problems using the parametrized point of view.
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