A Linear Kernel for Planar Total Dominating Set

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.Comment: 33 pages, 13 figure

Data Reductions and Combinatorial Bounds for Improved Approximation Algorithms

Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation algorithms. Our method features the use of approximation-preserving reductions, akin to the notion of parameterized reductions. We exemplify this method to obtain the currently best approximation algorithms for \textsc{Harmless Set}, \textsc{Differential} and \textsc{Multiple Nonblocker}, all of them can be considered in the context of securing networks or information propagation