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.

翻译：我们研究顶点加权图与网络中的最小权重支配集问题以及最小基数支配集的最小权重问题。后者是一个双目标优化问题，与经典的最小权重支配集问题不同——经典问题要求在图中找到权重最小的支配集，而不试图优化其基数。换言之，双目标问题中最小化支配集基数的目标可被视为约束条件，即寻找帕累托最优解的特例。首先，我们展示了如何通过整数线性规划公式将双目标优化问题转化为最小权重支配集问题。随后，在不同假设条件下，应用概率方法获得了图中最小权重支配集的上界。此外，还描述了在图中寻找小权重支配集的相应随机算法。最后，通过计算实验展示了针对两类不同随机图的结果。