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
Counting disjoint 2-partitions for points in the plane
Discrete Applied Mathematics
The partition bargaining problem
Discrete Applied Mathematics
Sphere-separable partitions of multi-parameter elements
Discrete Applied Mathematics
Algorithms for subset selection in linear regression
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On the number of separable partitions
Journal of Combinatorial Optimization
Discrete Optimization
Linear-shaped partition problems
Operations Research Letters
Are there more almost separable partitions than separable partitions?
Journal of Combinatorial Optimization
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We consider the class of shaped partition problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is fixed. In contrast, we show that when both d and p are fixed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of shaped partition polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is fixed; however, we show that when both d and p are fixed, the number of vertices of any shaped partition polytope is $O(n^{d{p\choose 2}})$ and all vertices can be produced in strongly polynomial time.