The vector partition problem for convex objective functions, S. Onn and L. J. Schulman. Mathematics of Operations Research, 26(3) 583-590, August 2001.

The {\em partition problem} concerns the partitioning of a given set of $n$ vectors in $d$-space into $p$ parts so as to maximize an objective function which is convex on the sum of vectors in each part. The problem has broad expressive power and captures NP-hard problems even if either $p$ or $d$ is fixed. In this article we show that when both $p,d$ are fixed, the problem is solvable in strongly polynomial time using $O(n^{d(p-1)-1})$ arithmetic operations. This improves upon the previously known bound of $O(n^{dp^2})$. Our method is based on the introduction of the {\em signing 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 called {\em partition polytope} of the same set of points.