Incremental Algorithm for Minimum Cut and Edge Connectivity in Hypergraph

For an uncapacitated hypergraph H = (V, E) with n = vertical bar V vertical bar, m = vertical bar E vertical bar and p = Sigma(e is an element of E)vertical bar e vertical bar, and edge connectivity lambda, this paper presents an insertion-only algorithm which updates minimum cut and edge connectivity incrementally on addition of a set of hyperedges to an existing hypergraph. The algorithm is deterministic and takes O(lambda n) amortized time per insertion of a hyperedge. The algorithm can answer queries on edge-connectivity in O(1) time and returns a cut of size lambda in O(n) time. First we propose a method to maintain a hypercactus [3] under the addition of a set of hyperedges. It is observed that the time for maintaining a hypercactus on addition of a set U of hyperdeges is O(n + p(u)) where p(u) = Sigma(e is an element of U)vertical bar e vertical bar. This method is then used as a subroutine in our incremental algorithm for maintaining minimum cut and edge connectivity.