The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is $W[P]$-complete when parameterized with respect to the solution size. We note that it was only known to be $W[2]$-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.

翻译：电力控制集（PDS）问题源于通过相量测量单元部署来监测电网的实际需求。该问题要求在图中选择最小顶点集，通过反复应用两条观测规则实现对所有剩余顶点的观测。我们的贡献包含两方面：首先，我们通过证明PDS在参数化解集规模下是$W[P]$-完备的，确定了其参数化复杂度——此前仅知其属于$W[2]$-困难问题；其次，也是主要贡献，我们提出了一种能高效求解实际算例的新算法。该算法由两个互补部分组成：第一部分是适用于PDS的归约规则集，可与现有算法协同使用；第二部分是基于隐式命中集方法求解剩余核的算法。在文献中的一组电网实例上进行的评估表明，我们的求解器平均性能比现有最优PDS求解器提升一个数量级以上。此外，该算法可在数分钟内求解此前无法处理的洲际规模算例。