Weighted domination models and randomized heuristics

TL;DR

This work studies minimum-weight and smallest-weight minimum-size dominating sets in vertex-weighted graphs, formulating a two-objective problem that can be reduced to a single-objective minimum-weight dominating set via an ILP-based weight transformation. It develops probabilistic upper bounds and three randomized heuristics (uniform, non-uniform, inverse) to construct small, light dominating sets, accompanied by an explicit algorithm to produce a minimal inclusion subset from a randomized start. The authors provide ILP reductions, derive generalized bounds extending classic Caro-Wei-type results to weighted graphs, and validate the approaches with extensive experiments on Erdős–Rényi and sun graphs, highlighting substantial scalability and practical performance benefits over generic ILP solvers on large instances. The results show that these simple, memory-efficient randomized methods can deliver competitive, sometimes superior, solutions quickly, making them useful for obtaining good initial solutions and guiding more expensive exact searches in practice. The work thus contributes both theoretical bounds for weighted domination and practical randomized tools for large-scale graph optimization problems.

Abstract

We consider the minimum weight and smallest weight minimum-size dominating set problems in vertex-weighted graphs and networks. The latter problem is a two-objective optimization problem, which is different from the classic minimum weight dominating set problem that requires finding a dominating set of the smallest weight in a graph without trying to optimize its cardinality. In other words, the objective of minimizing the size of the dominating set in the two-objective problem can be considered as a constraint, i.e. a particular case of finding Pareto-optimal solutions. First, we show how to reduce the two-objective optimization problem to the minimum weight dominating set problem by using Integer Linear Programming formulations. Then, under different assumptions, the probabilistic method is applied to obtain upper bounds on the minimum weight dominating sets in graphs. The corresponding randomized algorithms for finding small-weight dominating sets in graphs are described as well. Computational experiments are used to illustrate the results for two different types of random graphs.

Figures (6)

• Figure 1: Randomized small-weight dominating set
• Figure 2: Minimal Dominating Subset
• Figure 3: Performance profiles of the solvers for the Erdős--Rényi graph on $20,000$ vertices.
• Figure 4: Running the randomized heuristics on the Erdős--Rényi graph of $40,000$ vertices.
• Figure 5: Running the randomized heuristics on the sun graph of $10163$ vertices ($\delta=1250$).

Theorems & Definitions (6)

• Theorem 1: AS1992HHS1998
• Theorem 2: CR1997HHS1998HPV1999Lee1998
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6