Graph minors: X. obstructions to tree-decomposition
Journal of Combinatorial Theory Series B
Graph rewriting: an algebraic and logic approach
Handbook of theoretical computer science (vol. B)
How to allocate network centers
Journal of Algorithms
Approximation algorithms for NP-complete problems on planar graphs
Journal of the ACM (JACM)
Quickly excluding a planar graph
Journal of Combinatorial Theory Series B
Deciding first-order properties of locally tree-decomposable structures
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Approximation algorithms for independent sets in map graphs
Journal of Algorithms
Dominating sets in planar graphs: branch-width and exponential speed-up
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Parameterized Algorithms for Planar Dominating Set
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
Problems Easy for Tree-Decomposable Graphs (Extended Abstract)
ICALP '88 Proceedings of the 15th International Colloquium on Automata, Languages and Programming
Primal-Dual Algorithms for Connected Facility Location Problems
APPROX '02 Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization
TCS '02 Proceedings of the IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science: Foundations of Information Technology in the Era of Networking and Mobile Computing
Small k-Dominating Sets in Planar Graphs with Applications
WG '01 Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science
WG '02 Revised Papers from the 28th International Workshop on Graph-Theoretic Concepts in Computer Science
Exponential Speedup of Fixed-Parameter Algorithms on K3, 3-Minor-Free or K5-Minor-Free Graphs
ISAAC '02 Proceedings of the 13th International Symposium on Algorithms and Computation
Sequential and Parallel Algorithms for Embedding Problems on Classes of Partial k-Trees
SWAT '94 Proceedings of the 4th Scandinavian Workshop on Algorithm Theory
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Parameterized Complexity
Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Equivalence of local treewidth and linear local treewidth and its algorithmic applications
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Polynomial-time data reduction for dominating set
Journal of the ACM (JACM)
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
A refined search tree technique for Dominating Set on planar graphs
Journal of Computer and System Sciences
Packing and Covering δ-Hyperbolic Spaces by Balls
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Problems parameterized by treewidth tractable in single exponential time: a logical approach
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Generation of graphs with bounded branchwidth
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
New tools and simpler algorithms for branchwidth
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Fast algorithms for hard graph problems: bidimensionality, minors, and local treewidth
GD'04 Proceedings of the 12th international conference on Graph Drawing
Two birds with one stone: the best of branchwidth and treewidth with one algorithm
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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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 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 branch-width 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 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.