Implicit Hitting Set Enumeration for Power Dominating Sets

Overview & Synopsis

The Transversal Hypergraph problem asks to enumerate (list) all inclusion-minimal hitting sets of a hypergraph. It usually assumes that the input is completely known and does not change over the course of the enumeration. In many applications, the underlying hypergraph is too large or expensive to compute. This motivated the implicit hitting set framework [Chandrasekaran, Karp, Moreno-Centeno, and Vempala, SODA 2011] for the corresponding optimization problem of finding a smallest hitting set. Access to the input is given by an oracle that either confirms that a query set $S$ is a hitting set, or reveals a new hyperedge disjoint from $S$. We adapt this framework to enumeration. We show for a general class of algorithms that they are able to find all hitting sets when given access to such an oracle, even if the hypergraph is a priori unknown. We also prove that the current fastest Transversal Hypergraph algorithm, Minimal-to-Maximal Conversion Search by Murakami and Uno [DAM 2014], belongs to this class. We then engineer an implicit hitting set algorithm for the enumeration of minimal power dominating sets in graphs. A power dominating set is a selection of vertices that "observe" all vertices by exhaustively applying two rules: if a vertex is dominated by the selection, it becomes observed; and if an observed vertex has a unique unobserved neighbor, this neighbor becomes observed as well. An experimental evaluation on a collection of power grid instances from the literature shows that our approach achieves very high throughput rates.