The complexity of robot motion planning
The complexity of robot motion planning
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Constraint-based clustering in large databases
ICDT '01 Proceedings of the 8th International Conference on Database Theory
Clustering with Instance-level Constraints
ICML '00 Proceedings of the Seventeenth International Conference on Machine Learning
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
On the Complexity of Numerical Analysis
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
The Planar k-Means Problem is NP-Hard
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
NP-hardness of Euclidean sum-of-squares clustering
Machine Learning
Norm statistics and the complexity of clustering problems
Discrete Applied Mathematics
k-means requires exponentially many iterations even in the plane
Proceedings of the twenty-fifth annual symposium on Computational geometry
Data clustering with size constraints
Knowledge-Based Systems
Least squares quantization in PCM
IEEE Transactions on Information Theory
Multigraph realizations of degree sequences: Maximization is easy, minimization is hard
Operations Research Letters
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In this paper we study the complexity of some size constrained clustering problems with norm Lp. We obtain the following results: (i) A separation property for the constrained 2-clustering problem. This implies that the optimal solutions in the 1-dimensional case verify the so-called “String Property”; (ii) The NP-hardness of the constrained 2-clustering problem for every norm Lp (p 1); (iii) A polynomial time algorithm for the constrained 2-clustering problem in dimension 1 for every norm Lp with integer p. We also give evidence that this result cannot be extended to norm Lp with rational non-integer p; (iv) The NP-hardness of the constrained clustering problem in dimension 1 for every norm Lp (p ≥ 1).