TY - JOUR

T1 - Fixed-Parameter Algorithms for (k, r)-Center in Planar Graphs and Map Graphs

AU - Demaine, Erik D.

AU - Hajiaghayi, Mohammadtaghi

AU - Fomin, Fedor V.

AU - Thilikos, Dimitrios M.

AU - Thilikos, Dimitrios M.

PY - 2005

Y1 - 2005

N2 - 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 article, we prove that the (k, r)-center problem, parameterized by k and r, is fixed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity f (k, r)nO(1) where the function f is independent of n. In particular, we show that f (k, r) = 2O(r log r) √ k, 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 graphtheoretic 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 approach builds on the seminal results of Robertson and Seymour on Graph Minors, and as a result is much more powerful than the previous machinery of Alber et al. for exponential speedup on planar graphs. To demonstrate the versatility of our results, we show how our algorithms can be extended to general parameters that are “large” on grids. In addition, our use of branchwidth instead of the usual treewidth allows us to obtain much faster algorithms, and requires more complicated dynamic programming than the standard leaf/introduce/forget/join structure of nice tree decompositions. Our results are also unique in that they apply to classes of graphs that are not minor-closed, namely, constant powers of planar graphs and map graphs.

AB - 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 article, we prove that the (k, r)-center problem, parameterized by k and r, is fixed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity f (k, r)nO(1) where the function f is independent of n. In particular, we show that f (k, r) = 2O(r log r) √ k, 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 graphtheoretic 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 approach builds on the seminal results of Robertson and Seymour on Graph Minors, and as a result is much more powerful than the previous machinery of Alber et al. for exponential speedup on planar graphs. To demonstrate the versatility of our results, we show how our algorithms can be extended to general parameters that are “large” on grids. In addition, our use of branchwidth instead of the usual treewidth allows us to obtain much faster algorithms, and requires more complicated dynamic programming than the standard leaf/introduce/forget/join structure of nice tree decompositions. Our results are also unique in that they apply to classes of graphs that are not minor-closed, namely, constant powers of planar graphs and map graphs.

KW - (k, r)-center

KW - Algorithms

KW - Domination

KW - Fixed-Parameter algorithms

KW - Graph

KW - Map graph

KW - Planar

KW - Theory

UR - http://www.scopus.com/inward/record.url?scp=84948770618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948770618&partnerID=8YFLogxK

U2 - 10.1145/1077464.1077468

DO - 10.1145/1077464.1077468

M3 - Article

AN - SCOPUS:84948770618

SN - 1549-6325

VL - 1

SP - 33

EP - 47

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 1

ER -