Parameterized Approximation Algorithms for some Location Problems in Graphs

We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) $r$-Domination problem and the (Connected) $p$-Center problem for unweighted and undirected graphs. Given a graph $G$, we show how to construct a (connected) $\big(r + \mathcal{O}(μ) \big)$-dominating set $D$ with $|D| \leq |D^*|$ efficiently. Here, $D^*$ is a minimum (connected) $r$-dominating set of $G$ and $μ$ is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of $G$. Additionally, we show that a $+ \mathcal{O}(μ)$-approximation for the (Connected) $p$-Center problem on $G$ can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.