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

TL;DR

This work addresses the problem of enumerating the $k$-best minimal connected edge dominating sets under a cardinality constraint, a task that is typically hard when solved exactly. It introduces an approximation-enumeration framework and leverages a supergraph $\mathcal{G}$ of minimal connected edge dominating sets, with three neighbor types guiding transitions and a linear-time $\mu(\cdot)$ procedure to maintain minimality. The authors prove strong connectivity of $\mathcal{G}$ and achieve polynomial-delay enumeration of all minimal sets, then extend to $k$-best enumeration with a constant-factor approximation: a $4$-approximation for the $k$-best problem, given a $2$-approximation for the base minimum problem, and delay $O(n m^2 \Delta)$. This yields scalable, approximate top-$k$ solutions for connected edge domination under cardinality constraints, with practical implications for graph-structured data analyses.

Abstract

\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 $\order{nm^2Δ}$ delay, where $n$, $m$, $Δ$ are the number of vertices, the number of edges, and the maximum degree of an input graph.

Figures (2)

• Figure 1: An example of a trivial case. The left graph has a minimum connected edge dominating set with cardinality one. The type-II solutions are three minimal solutions with two edges represented by dotted edges, gray edges, and black edges in the middle graph. The type-III solution is one solution with three edges represented by gray edges in the right graph.
• Figure 2: Examples for three types of neighbors of a minimal connected edge dominating set $X$. (a) depicts $X$. Black bold lines indicate the edges in $X$. (b) depicts a type-I neighbor $Y_1$ of $X$. $Y_1$ is obtained by removing a dashed edge corresponding $e$ and adding a wavy path corresponding $\{f, g\}$. The bottom edge is removed by applying $\mu(\cdot)$. (c) and (d) are a type-II and a type-III neighbor of $X$, respectively. Similarly, the neighbors are obtained by removing a dashed edge and adding wavy edges.

Theorems & Definitions (30)

• theorem 1
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