Polynomial Delay and Space Discovery of Connected and Acyclic Sub-hypergraphs in a Hypergraph

Abstract

In this paper, we study the problem of finding all tree-like substructure contained in a hypergraph, with potential applications to substructure mining from relational data. We employ the class of connected and Berge acyclic sub-hypergraphs as definition of tree-like substructures, which is the most restricted notion of acyclicities for hypergraphs. Then, we present an efficient depth-first algorithm that finds all connected and Berge acyclic sub-hypergraphs S in a hypergraph $\mathcal H$ with m hyperedges and n vertices in O(nm ²) time per solution (delay) using O(N) space, where $N = ||\mathcal H||$ is the total input size. To achieve efficient enumeration, we use the notion of the maximum border set. This result gives the first polynomial delay and time algorithm for enumeration of connected and Berge-acyclic sub-hypergraphs. We also present an incremental enumeration algorithm that finds all solutions S in $O(\varDelta MB(S)\tau(m)) = O(rd\cdot\tau(m))$ delay using O(N) space and preprocessing, whose delay depends only on the difference of solutions, where S is the enumerated sub-hypergraph, $\varDelta MB(S)$ is the number of newly added hyperedges to the maximum border of S, r and d are the rank and degree of $\mathcal H$, respectively, and τ(m) = ((log log m)²/log log log m).