Erratum: generalized selection and ranking: sorted matrices
SIAM Journal on Computing
On the complexity of some geometric problems in unbounded dimension
Journal of Symbolic Computation
Rectilinear and polygonal p-piercing and p-center problems
Proceedings of the twelfth annual symposium on Computational geometry
Rectilinear p-piercing problems
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Efficient algorithms for geometric optimization
ACM Computing Surveys (CSUR)
Faster construction of planar two-centers
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
More planar two-center algorithms
Computational Geometry: Theory and Applications
Which problems have strongly exponential complexity?
Journal of Computer and System Sciences
Linear FPT reductions and computational lower bounds
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Fixed-parameter algorithms for (k, r)-center in planar graphs and map graphs
ACM Transactions on Algorithms (TALG)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Tight lower bounds for certain parameterized NP-hard problems
Information and Computation
Efficient approximation schemes for geometric problems?
ESA'05 Proceedings of the 13th annual European conference on Algorithms
The complexity of geometric problems in high dimension
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
Hardness of discrepancy computation and ε-net verification in high dimension
Journal of Complexity
Hi-index | 0.00 |
We study the parameterized complexity of the k-center problem on a given n-point set P in &ℝd, with the dimension d as the parameter. We show that the rectilinear 3-center problem is fixed-parameter tractable, by giving an algorithm that runs in O(n log n) time for any fixed dimension d. On the other hand, we show that this is unlikely to be the case with both the Euclidean and rectilinear k-center problems for any k≥ 2 and k ≥ 4 respectively. In particular, we prove that deciding whether P can be covered by the union of 2 balls of given radius or by the union of 4 cubes of given side length is W[1]-hard with respect to d, and thus not fixed-parameter tractable unless FPT=W[1]. For the Euclidean case, we also show that even an no(d)-time algorithm does not exist, unless there is a 2o(n)-time algorithm for n-variable 3SAT, that is, the Exponential Time Hypothesis fails.