An Approximation Algorithm for $K$-best Enumeration of Minimal Connected Edge Dominating Sets with Cardinality Constraints

\emph{$K$-best enumeration}, which asks to output $k$ best solutions without duplication, plays an important role in data analysis for many fields. In such fields, data can be typically represented by graphs, and thus subgraph enumeration has been paid much attention to. However, $k$-best enumeration tends to be intractable since, in many cases, finding one optimum solution is \NP-hard. To overcome this difficulty, we combine $k$-best enumeration with a new concept of enumeration algorithms called \emph{approximation enumeration algorithms}, which has been recently proposed. As a main result, we propose an $\alpha$-approximation algorithm for minimal connected edge dominating sets which outputs $k$ minimal solutions with cardinality at most $\alpha\cdot\overline{\rm OPT}$, where $\overline{\rm OPT}$ is the cardinality of a mini\emph{mum} solution which is \emph{not} outputted by the algorithm, and $\alpha$ is constant. Moreover, our proposed algorithm runs in $O(nm^2\Delta)$ delay, where $n$, $m$, $\Delta$ are the number of vertices, the number of edges, and the maximum degree of an input graph.