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

TL;DR

This paper introduces a 4-approximation algorithm for efficiently enumerating minimal connected edge dominating sets with bounded size, addressing the intractability of exact solutions in large graphs.

Contribution

It presents the first approximation enumeration algorithm for minimal connected edge dominating sets with a guaranteed approximation ratio and polynomial delay.

Findings

Outputs up to 4 times the optimal solution size

Runs in polynomial delay of order nm^2Δ

Provides a practical approach for large graph enumeration

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