  \emph{$K$-best enumeration}, which asks to output $k$-best solutions without
duplication, is a helpful tool in data analysis for many fields. In such
fields, graphs typically represent data. Thus subgraph enumeration has been
paid much attention to such fields. 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 concept of
enumeration algorithms called \emph{approximation enumeration algorithms}. As a
main result, we propose a $4$-approximation algorithm for minimal connected
edge dominating sets which outputs $k$ minimal solutions with cardinality at
most $4\cdot\OPT$, where $\OPT$ is the cardinality of a minimum solution which
is \emph{not} outputted by the algorithm. Our proposed algorithm runs in
$\order{nm^2\Delta}$ delay, where $n$, $m$, $\Delta$ are the number of
vertices, the number of edges, and the maximum degree of an input graph.
