Optimal coteries for rings and related networks

SummaryLet a distributed system be represented by a graphG=(V, E), whereV is the set of nodes andE is the set of communication links. A coterie is defined as a family,C, of subsets ofV such that any pair of subsets inC has at least one node in common and no subset inC contains any other subset inC. Assuming that each nodevi∈V (resp. linkej∈E) is operational with probabilitypi (resp.rj), the availability of a coterie is defined as the probability that the operational nodes and links ofG connect all nodes in at least one subset in the coterie. Although it is computationally intractable to find an optimal coterie that maximizes availability for general graphG, we show in this paper that, ifG is a ring, either a singleton coterie or a 3-majority coterie is optimal. Therefore, for any ring, an optimal coterie can be computed in polynomial time. This result is extended to the more general graphs, in which each biconnected component is either an edge or a ring.