 
Summary: The vector partition problem for convex objective
functions
Shmuel Onn # Leonard J. Schulman +
Abstract
The partition problem concerns the partitioning of a given set of n vectors in dspace 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 NPhard 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(p1)1 ) arithmetic operations. This improves upon the
previously known bound of O(n dp 2
). Our method is based on the introduction of the 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 partition polytope of the same set of points.
Mathematics Subject Classifications: 05A, 15A, 51M, 52A, 52B, 52C, 68Q, 68R, 68U,
90B, 90C
Keywords: partition, cluster, optimization, separation, convex, polytope, zonotope, sign
ing, vertex enumeration, polynomial time, combinatorial optimization
1 Introduction
The partition problem concerns the partitioning of a set A = {a 1 , . . . , a n } of n vectors in dspace
into p parts so as to maximize an objective function which is convex on the sum of vectors in
