Domination problems in general can capture situations in which some entities have an effect on other entities (and sometimes on themselves). The usual goal is to select a minimum number of entities that can influence a target group of entities or to influence a maximum number of target entities with a certain number of available influencers. In this work, we focus on the distinction between \textit{internal} and \textit{external} domination in the respective maximization problem. In particular, a dominator can dominate its entire neighborhood in a graph, internally dominating itself, while those of its neighbors which are not dominators themselves are externally dominated. We study the problem of maximizing the external domination that a given number of dominators can yield and we present a 0.5307-approximation algorithm for this problem. Moreover, our methods provide a framework for approximating a number of problems that can be cast in terms of external domination. In particular, we observe that an interesting interpretation of the maximum coverage problem can capture a new problem in elections, in which we want to maximize the number of \textit{externally represented} voters. We study this problem in two different settings, namely Non-Secrecy and Rational-Candidate, and provide approximability analysis for two alternative approaches; our analysis reveals, among other contributions, that an earlier resource allocation algorithm is, in fact, a 0.462-approximation algorithm for maximum external domination in directed graphs.

翻译：支配问题通常可以刻画某些实体对其他实体（有时包括自身）产生影响的情境。常见目标是选择最少数量的实体以影响目标群体，或利用固定数量的影响者最大化目标实体的影响范围。本研究聚焦于各自最大化问题中“内部支配”与“外部支配”的区分。具体而言，一个支配者可以支配其所在图中的完整邻域（即内部支配自身），而未被选为支配者的邻居节点则受其外部支配。我们研究如何通过给定数量的支配者最大化外部支配效果，并提出一种0.5307近似算法。此外，我们的方法为可基于外部支配概念建模的若干问题提供了近似框架。特别地，我们发现最大覆盖问题的一种有趣解释可衍生选举领域的新问题：即最大化“外部代表”选民的数量。我们在两种不同场景（非保密场景与理性候选人场景）中研究该问题，并为两种替代方法提供可近似性分析；分析结果表明（除其他贡献外），早期的一种资源分配算法实际上可视为有向图中最大外部支配问题的0.462近似算法。