加权支配模型与随机化启发式算法
Weighted domination models and randomized heuristics
AI总结：
本文研究加权图中的最小权及最小权最小规模支配集问题，通过整数线性规划归约双目标模型，并用概率方法给出上界与随机算法，在两类随机图上验证。
摘要：
我们研究顶点加权图与网络中的最小权问题以及最小权最小规模支配集问题。后一问题是一个双目标优化问题，不同于经典的最小权支配集问题：后者要求在图中寻找权值最小的支配集，而不尝试优化其基数。换言之，在该双目标问题中，最小化支配集规模这一目标可被视为一个约束，即寻找 Pareto 最优解的一种特殊情形。首先，我们展示如何利用整数线性规划公式，将该双目标优化问题归约为最小权支配集问题。随后，在不同假设下，应用概率方法获得图中最小权支配集的上界。文中还描述了用于在图中寻找小权值支配集的相应随机化算法。计算实验用来说明两类不同随机图上的结果。
英文摘要：
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.
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