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
Graphs of transportation polytopes
Journal of Combinatorial Theory Series A
On the number of separable partitions
Journal of Combinatorial Optimization
Discrete Optimization
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Thepartition problem concerns the partitioning of a given set ofn vectors ind-space intop parts to maximize an objective function that is convex on the sum of vectors in each part. The problem has broad expressive power and captures NP-hard problems even if eitherp ord is fixed. In this article we show that when bothp,d are fixed, the problem is solvable in strongly polynomial time usingO(n d(p-1)-1) arithmetic operations. This improves upon the previously known bound ofO( ndp 2 ). Our method is based on the introduction of thesigning zonotope of a set of points in space. We study this object, which is of interest in its own right, and show that it is a refinement of the so-calledpartition polytope of the same set of points.