Fixed-parameter algorithms for the (k, r)-center in planar graphs and map graphs

The (k; r)-center problem asks whether an input graph G has k vertices (called centers) such that every vertex of G is within distance r from some center. In this paper we prove that the (k; r)-center problem, parameterized by k and r, is xed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity f(k; r)n where the function f is independent of n. In particular, we show that f(k; r) = 2 O(r log r) , where the exponent of the exponential term grows sublinearly in the number of centers. Moreover, we prove that the same type of FPT algorithms can be designed for the more general class of map graphs introduced by Chen, Grigni, and Papadimitriou. Our results combine dynamic-programming algorithms for graphs of small branchwidth and a graph-theoretic result bounding this parameter in terms of k and r. Finally, a byproduct of our algorithm is the existence of a PTAS for the r-domination problem in both planar graphs and map graphs. Our approac