Momentopes, the Complexity of Vector Partitioning, and Davenport—Schinzel Sequences
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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 ν p,d (n)= Ω(n ⌊(d−1)/2⌋M 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 ν 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.