Weighted domination models and randomized heuristics

TL;DR

This paper explores weighted domination problems in graphs, reducing a two-objective problem to a single-objective ILP, and introduces randomized heuristics with computational validation on random graph types.

Contribution

It presents a reduction of a two-objective domination problem to ILP and develops randomized algorithms with theoretical bounds and experimental validation.

Findings

Upper bounds on minimum weight dominating sets derived using probabilistic methods

Randomized heuristics effectively find small-weight dominating sets

Computational experiments demonstrate the algorithms' performance on random graphs

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.