On clustering problems with connected optima in Euclidean spaces
Discrete Mathematics
Optimal partitions having disjoint convex and conic hulls
Mathematical Programming: Series A and B
Enumerating nested and consecutive partitions
Journal of Combinatorial Theory Series A
Directional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions
Mathematics of Operations Research
Discrete Applied Mathematics
The Vector Partition Problem for Convex Objective Functions
Mathematics of Operations Research
Optimality of Nested Partitions and Its Application to Cluster Analysis
SIAM Journal on Optimization
A Polynomial Time Algorithm for Shaped Partition Problems
SIAM Journal on Optimization
Vertex characterization of partition polytopes of bipartitions and of planar point sets
Discrete Applied Mathematics - Workshop on discrete optimization DO'99, contributions to discrete optimization
Sortability of vector partitions
Discrete Mathematics
Approximation algorithms for the metric maximum clustering problem with given cluster sizes
Operations Research Letters
On the number of separable partitions
Journal of Combinatorial Optimization
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We show the optimality of sphere-separable partitions for problems where n vectors in d-dimensional space are to be partitioned into p categories to minimize a cost function which is dependent in the sum of the vectors in each category; the sum of the squares of their Euclidean norms; and the number of elements in each category. We further show that the number of these partitions is polynomial in n. These results broaden the class of partition problems for which an optimal solution is guaranteed within a prescribed set whose size is polynomially bounded in n. Applications of the results are demonstrated through examples.