The Complexity of Three-Way Statistical Tables

Multi-way tables with specified marginals arise in a variety of applications in statistics and operations research. We provide a comprehensive complexity classification of three fundamental computational problems on tables: existence, counting and entry-security. One major outcome of our work is that each of the following problems is intractable already for "slim" 3-tables, with constant and smallest possible number 3 of rows: (1) deciding existence of 3-tables with given consistent 2-marginals; (2) counting all 3-tables with given 2-marginals; (3) finding whether an integer value is attained in entry (i,j,k) by at least one of the 3-tables satisfying given (feasible) 2-marginals. This implies that a characterization of feasible marginals for such slim tables, sought by much recent research, is unlikely to exist. Another important consequence of our study is a systematic efficient way of embedding the set of 3-tables satisfying any given 1-marginals and entry upper bounds in a set of slim 3-tables satisfying suitable 2-marginals with no entry bounds. This provides a valuable tool for studying multi-index transportation problems and multi-index transportation polytopes

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 vp,d(n) = �(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 vp,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