Algorithms for Minimum Membership Dominating Set Problem

TL;DR

The paper studies the Minimum Membership Dominating Set problem, requiring that every vertex has between 1 and $k$ neighbors in a chosen set $S$. It delivers an exact exponential algorithm for split graphs with runtime $O^*(1.747^n)$, proves ETH-based hardness for bipartite graphs with $k\ge2$, and shows NP-hardness under $\Delta = k+2$ for $k\ge5$ in bipartite graphs. It further develops FPT algorithms for structural parameters (twin cover and distance to cluster) and provides a linear-time DP for trees. Collectively, these results advance understanding of MMDS complexity, offering efficient algorithms and kernelization insights across graph classes and structural parameters, with potential implications for related domination-type problems.

Abstract

Given a graph $G = (V, E)$ and an integer $k$, the Minimum Membership Dominating Set problem asks to compute a set $S \subseteq V$ such that for each $v \in V$, $1 \leq |N[v] \cap S| \leq k$. The problem is known to be NP-complete even on split graphs and planar bipartite graphs. In this paper, we approach the problem from the algorithmic standpoint and obtain several interesting results. We give an $\mathcal{O}^*(1.747^n)$ time algorithm for the problem on split graphs. Following a reduction from a special case of 1-in-3 SAT problem, we show that there is no sub-exponential time algorithm running in time $\mathcal{O}^*(2^{o(n)})$ for bipartite graphs, for any $k \geq 2$. We also prove that the problem is NP-complete when $Δ= k+2$, for any $k\geq 5$, even for bipartite graphs. We investigate the parameterized complexity of the problem for the parameter twin cover and the combined parameter distance to cluster, membership($k$) and prove that the problem is fixed-parameter tractable. Using a dynamic programming based approach, we obtain a linear-time algorithm for trees.

Figures (6)

• Figure 1: Posing a section of the MMDS problem as an instance of the Set Cover
• Figure 2: (a) Variable gadget for some $i \in [n]$ (on the left). (b) Literal gadget for some $i \in [n]$ (on the right).
• Figure 3: Construction of an MMDS instance from the 3-CNF$^{\leq 3}$-XSAT formula: $\phi = (x_1 \vee \bar{x_2} \vee x_3) \wedge (\bar{x_1} \vee \bar{x_2} \vee x_3) \wedge (\bar{x_1} \vee x_2 \vee \bar{x_3})$. Here, the red circles and blue rectangles are placeholders for literal gadget and variable gadgets. The literal gadget and variable gadget are illustrated in \ref{['fig: Fig4']}
• Figure 4: Partitioning of the vertex set $V$ into sets $T$ and $C$, where $T$ is the twin cover and $C$ is the union of clique sets outside $T$. Square shaped vertices are a part of $S$.
• Figure 5: Partitioning of the vertex set $V$ into sets $D$ and $C$, where $|D|$ is the distance to cluster and $C$ is the union of clique sets outside $D$. The collective neighbourhood of $C_3$ in $D$ is shown.