Decremental Maintenance of Reachability in Hypergraphs and Minimum Models of Horn Formulae

In this paper we present a decremental algorithm for maintaining of minimum rank hyperpaths in a directed hypergraph from a source vertex s to all other vertices, under the assumption of unit hyperedge weights. Given a hypergraph H with n vertices and m hyperedges, the total time needed to perform a sequence of m hyperedge deletions is O(n Size(H)), where Size(H) is the sum of the sizes of the hyperedges of H; the total space needed is O(n + Size(H)). Thus, the amortized time per deletion is O (Size(H) n=m). In the case of integer hyperedge weights in 1; C] our solution requires O(CnSize(H)) total time and O(C + n + Size(H)) space. The only algorithm proposed in the literature for the fully dynamic maintenance of minimum weight hyperpaths may take total time O(n 2 Size(H)) for a sequence of m deletions. Thus, our algorithm gives an n speedup for the amortized time per deletion. Using the algorithm presented in this paper, we also show how to maintain the satissability and the minimum model of a Horn formula F in total time O(n Length(F)) over a sequence of m clause deletions. Actually, the algorithm does more, since for each propositional symbol in the formula it also maintains the depth of its shortest proof.