Parameterized Complexity of Minimum Membership Dominating Set

TL;DR

This paper explores the parameterized complexity of the Minimum Membership Dominating Set problem, revealing its computational hardness and providing an optimal algorithm based on vertex cover size.

Contribution

It establishes W[1]-hardness results for MMDS under various parameters and presents an optimal algorithm parameterized by vertex cover size.

Findings

W[1]-hardness with respect to pathwidth and treewidth.

W[1]-hardness on split graphs when parameterized by k.

An optimal algorithm running in 2^{O(vc)}|V|^{O(1)} time.

Abstract

Given a graph $G=(V,E)$ and an integer $k$, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set $S \subseteq V$ of $G$ such that for each $v \in V$, $|N[v] \cap S|$ is at most $k$. We investigate the parameterized complexity of the problem and obtain the following results about MMDS: W[1]-hardness of the problem parameterized by the pathwidth (and thus, treewidth) of the input graph. W[1]-hardness parameterized by $k$ on split graphs. An algorithm running in time $2^{\mathcal{O}(\textbf{vc})} |V|^{\mathcal{O}(1)}$, where $\textbf{vc}$ is the size of a minimum-sized vertex cover of the input graph. An ETH-based lower bound showing that the algorithm mentioned in the previous item is optimal.