Momentopes, the complexity of vector partitioning, and davenport-schinzel sequences

The computational complexity of the partition problem, which concerns the partitioning of a 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, is determined by the number of vertices of the corresponding p-partition polytope defined to be the convex hull in (d × p)-space of all solutions. In this article, introducing and using the class of Momentopes, we establish the lower bound v_p,d(n) = ω(n^[(d-1)/2]p) on the maximum number of vertices of any p-partition polytope of a set of n points in d-space, which is quite compatible with the recent upper bound v_p,d(n) = O(n^d(p-1)-1), implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability of Davenport-Schinzel sequences and describe some further properties of Momentopes.