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Reducing Dominating Sets in Graphs

Ernesto Parra Inza, José María Sigarreta Almira, Nodari Vakhania

TL;DR

The paper addresses the dominating set problem by enhancing a two-stage heuristic: a fast greedy Stage 1 to generate an initial dominating set, followed by a purification Stage 2 aimed at reducing it. It introduces four purification procedures (PP$_1$–PP$_4$) built on a width-based clustering of the initial set, leveraging private neighborhoods and inner/outer cover notions to guide vertex removal, and shows these procedures outperform the previous PP$_0$. Theoretical results provide a piecewise approximation bound for the purification stage, complemented by empirical evidence on over a thousand benchmark instances, where the methods achieve substantial improvements over existing upper bounds and exhibit notable optimality on known-optimum instances. The approach demonstrates strong practical scalability, with fast runtimes on large graphs and meaningful reductions in solution sizes, contributing to more efficient domination-based modeling and applications.

Abstract

A dominating set of a graph $G=(V,E)$ is a subset of vertices $S\subseteq V$ such that every vertex $v\in V\setminus S$ has at least one neighbor in set $S$. The corresponding optimization problem is known to be NP-hard. The best known polynomial time approximation algorithm for the problem separates the solution process in two stages applying first a fast greedy algorithm to obtain an initial dominating set, and then it uses an iterative procedure to reduce (purify) this dominating set. The purification stage turned out to be practically efficient. Here we further strengthen the purification stage presenting four new purification algorithms. All four purification procedures outperform the earlier purification procedure. The algorithms were tested for over 1300 benchmark problem instances. Compared to the known upper bounds, the obtained solutions were about 7 times better. Remarkably, for the 500 benchmark instances for which the optimum is known, the optimal solutions were obtained for 46.33\% of the tested instances, whereas the average error for the remaining instances was about 1.01.

Reducing Dominating Sets in Graphs

TL;DR

The paper addresses the dominating set problem by enhancing a two-stage heuristic: a fast greedy Stage 1 to generate an initial dominating set, followed by a purification Stage 2 aimed at reducing it. It introduces four purification procedures (PP–PP) built on a width-based clustering of the initial set, leveraging private neighborhoods and inner/outer cover notions to guide vertex removal, and shows these procedures outperform the previous PP. Theoretical results provide a piecewise approximation bound for the purification stage, complemented by empirical evidence on over a thousand benchmark instances, where the methods achieve substantial improvements over existing upper bounds and exhibit notable optimality on known-optimum instances. The approach demonstrates strong practical scalability, with fast runtimes on large graphs and meaningful reductions in solution sizes, contributing to more efficient domination-based modeling and applications.

Abstract

A dominating set of a graph is a subset of vertices such that every vertex has at least one neighbor in set . The corresponding optimization problem is known to be NP-hard. The best known polynomial time approximation algorithm for the problem separates the solution process in two stages applying first a fast greedy algorithm to obtain an initial dominating set, and then it uses an iterative procedure to reduce (purify) this dominating set. The purification stage turned out to be practically efficient. Here we further strengthen the purification stage presenting four new purification algorithms. All four purification procedures outperform the earlier purification procedure. The algorithms were tested for over 1300 benchmark problem instances. Compared to the known upper bounds, the obtained solutions were about 7 times better. Remarkably, for the 500 benchmark instances for which the optimum is known, the optimal solutions were obtained for 46.33\% of the tested instances, whereas the average error for the remaining instances was about 1.01.
Paper Structure (6 sections, 1 theorem, 1 equation, 4 figures)

This paper contains 6 sections, 1 theorem, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stage 1: The clustering
  4. Stage 2: Purification procedures
  5. Approximation ratios
  6. Experimental results

Key Result

Theorem 1

Stage 1 runs in time $O(n^3)$.

Figures (4)

  • Figure 1: Forests $T^{h-1}$ and $T^{h}$ (the red edge in $T^{h-1}$ is omitted in $T^{h}$
  • Figure 2: Results of purification procedures ($\min\{S^*_i\}$), $\gamma(G)$, and $U$
  • Figure 3: Time of purification procedures
  • Figure 4: Problem instance density and size vs the corresponding maximum execution time

Theorems & Definitions (2)

  • Theorem 1
  • Proof 1