Revisiting a Successful Reduction Rule for Dominating Set

TL;DR

This work delivers the first O(n+m)-time algorithm for applying Rule 1 in the Dominating Set problem on general graphs, showing that a single application suffices to realize the full reduction. By introducing canonical reference nodes and a linear-time pipeline to compute all suitable references and witnesses, the authors provide a practical framework that extends Rule 1 with additional pruning strategies. Empirically, the approach achieves significant data reduction and runtime speedups across real-world and synthetic graphs, with Plus/Extra variants removing orders of magnitude more nodes and edges than the original formulation and substantially improving Greedy solutions. The results demonstrate the practicality and impact of advanced reduction rules in preprocessing for Dominating Set, offering a foundation for more powerful kernelization and scalable solvers.

Abstract

Given a graph $G = (V, E)$ with $n$ vertices and $m$ edges, the DominatingSet problem asks for a set $D \subseteq V$ of minimal cardinality such that every vertex either is in $D$ or adjacent to a member of $D$. Although there is little hope for a kernelization algorithm on general graphs due to the W[2]-hardness of DominatingSet, data reduction rules are extensively used in practice. In this context, Rule1 due to Alber, Fellows, and Niedermeier [JACM 2004] has been shown to be very powerful, yet its best-known running time is $\mathcal{O}(n^3)$ ($= \mathcal{O}(nm)$) for general graphs. In this work, we propose, to the best of our knowledge, the first $\mathcal{O}(n + m)$-time algorithm for Rule1 on general graphs. We additionally propose simple, but practically significant, extensions to our algorithmic framework to further prune the input instances. We complement our theoretical claims with experiments that confirm the practicality of our approach. On average, we see significant speedups of over one order of magnitude while removing $59.8\times$ more nodes and $410.9\times$ more edges than the original formulation across a large dataset comprised of real-world and synthetic networks.

Figures (8)

• Figure 3.1: Neighbor types relative to reference node ${\color{ref}\rho}\xspace$. As nodes $a$ and $b$ have neighbors outside of $N[{\color{ref}\rho}\xspace]$, they are of type 1. The remaining type 2 and 3 have only neighbors within $N[{\color{ref}\rho}\xspace]$, with the distinction that nodes $c$ and $d$ are connected to type 1 nodes and thus are type 2. Since ${\color{ref}\rho}\xspace$ has at least one type 3 node, Rule 1 applies: We add $\rho$ to the dominating set. Then, it covers all its neighbors and there is no reason to add type 2 and 3 nodes into the dominating set. Hence, we can safely delete them. Observe that we keep type 1 nodes, since they may be needed to cover the "outside" nodes.
• Figure 3.2: Node $v$ appears as different neighbor types for different reference nodes, namely: $v \in N_1(\rho_1)$ as $\rho_1' \notin N(\rho_1)$. Furthermore, we have $v \in N_1(\rho'_1)$ as ${\rho_1 \notin N(\rho'_1)}$. Furthermore, $v \in N_2(\rho_2)$ as $N[v] \subseteq N[\rho_2]$ and $v$ is connected to $\rho_3 \in N_1(\rho_2)$ which is a type 1 neighbor to $\rho_2$. Finally, we also have $v \in N_3(\rho_3)$ as $N[v] \subseteq N[\rho_3]$ and $N(v) \cap N_1(\rho_3) = \emptyset$.
• Figure 4.1: Super-linear runtime for the naïve Rule 1 for the pre-selected reference nodes ${\color{ref}\mathsf{ref}}(S23\xspace)$. Construction: The two subgraphs $L$ and $R$ are $k$-cliques on nodes $V_L = \left\{\ell_1, \ldots, \ell_k\right\}$ and $V_R = \left\{r_1, \ldots, r_k\right\}$, respectively. Each $\ell_i$ is connected to all nodes in $V_R$. Further, we attach to $r_i \in V_R$ its own copy of the gadget illustrated. Observe that (i) $a_i \in N_1(r_i)$, (ii) $b_i \in N_2(r_i)$, and (iii) that the whole gadget cannot be reduced by Rule 1. We assume that the graph is provided as an ordered adjacency list, where $\ell_i$ are listed before $r_j$ and $a_i, b_i$. Analysis: Let $k > 1$. Then, each gadget's $b_i \in N_2(r_i)$ will pick $r_i$ as its canonical reference node, i.e., $\left\{({\color{wit}{b_i}} \lhd_{\left\{2,3\right\}} {\color{ref}r_i})\;\middle|\;1 \le i \le k\right\} \subseteq S23\xspace$ while the graph has no type 3 nodes at all (nodes in $L$ and $R$ will also pick a reference node in $R$). Now in order to partition the neighborhood $N(r_i)$ for a fixed $r_i \in R$, we have to inspect the neighborhood of each node in $V_L$ which takes time $\Theta(k^2)$. Thus, in total the naïve test requires $\Omega(k^3)=\Omega(n^3)$ time, as $n = \Theta(k)$ and $m=\Theta(k^2)$.
• Figure 5.1: Deletion of type 1 nodes. Rule 1 applies to $\mathcal{R}\xspace = \left\{{\color{ref}\rho}\xspace, {\color{ref}\sigma}\xspace\right\}$ as witnessed by $x \in N_3({\color{ref}\rho}\xspace)$ and $y \in N_3({\color{ref}\sigma}\xspace)$. Observe that the middle nodes ${u}\xspace$ and ${v}\xspace$ are type 1 neighbors to ${\color{ref}\rho}\xspace$ and ${\color{ref}\sigma}\xspace$, respectively. Hence, the original Rule 1 leaves them untouched. In contrast, \ref{['alg:remove']} marks all nodes. Thus it also deletes the middle nodes.
• Figure 5.2: Deletion of more type 1 nodes. Rule 1 applies to $\mathcal{R}\xspace = \left\{{\color{ref}\rho}\xspace, {\color{ref}\sigma}\xspace\right\}$ as witnessed by $x \in N_3({\color{ref}\rho}\xspace)$ and $y \in N_3({\color{ref}\sigma}\xspace)$. Then, \ref{['alg:remove']} marks all nodes but the middle $c$ which prevents the deletion of nodes ${u}\xspace$ and ${v}\xspace$ (since they are adjacent to the unmarked $c$).

Theorems & Definitions (19)

• Lemma 3.1
• Proof 1
• Lemma 3.2
• Proof 2
• Lemma 3.3
• Proof 3
• Lemma 4.1
• Proof 4
• Definition 4.2
• Lemma 4.3