Shmuel Onn

Markov bases of three-way tables are arbitrarily complicated

We show the following two universality statements on the entry-ranges and Markov bases of spaces of 3-way contingency tables with fixed 2-margins: (1) For any finite set D of nonnegative integers, there are r,c, and 2-margins for (r,c,3)-tables such that the set of values occurring in a fixed entry in all possible tables with these margins is D. (2) For any integer n-vector d, there are r,c such that any Markov basis for (r,c,3)-tables with fixed 2-margins must contain an element whose restriction to some n entries is d. In particular, the degree and support of elements in the minimal Markov bases when r and c vary can be arbitrarily large, in striking contrast with the case for 1-margined tables in any dimension and any format and with 2-margined (r,c,h)-tables with both c,h fixed. These results have implications for confidential statistical data disclosure control. Specifically, they demonstrate that the entry-range of 2-margined 3-tables can contain arbitrary gaps, suggesting that even if the smallest and largest possible values of an entry are far apart, the disclosure of such margins may be insecure. Thus, the behavior of sensitive data under disclosure of aggregated data is far from what has been so far believed. Our results therefore call for the re-examination of aggregation and disclosure practices and for further research on the issues exposed herein. Our constructions also provides a powerful automatic tool in constructing concrete examples, such as the possibly smallest 2-margins for (6, 4, 3)-tables with entry-range containing a gap.

N-fold integer programming

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed n-fold integer programming problems are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and the so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the d-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting-stock problem.

Convex Combinatorial Optimization

Abstract We introduce the convex combinatorial optimization problem, a far-reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

A Polynomial Time Algorithm for Shaped Partition Problems

We consider the class of shaped partition problems of partitioning n given vectors in d-dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part, subject to arbitrary constraints on the number of elements in each part. This class has broad expressive power and captures NP-hard problems even if either d or p is fixed. In contrast, we show that when both d and p are fixed, the problem can be solved in strongly polynomial time. Our solution method relies on studying the corresponding class of shaped partition polytopes. Such polytopes may have exponentially many vertices and facets even when one of d or p is fixed; however, we show that when both d and p are fixed, the number of vertices of any shaped partition polytope is

Colourful Linear Programming and its Relatives

O(n^{d{p\choose 2}})

All Linear and Integer Programs Are Slim 3-Way Transportation Programs

and all vertices can be produced in strongly polynomial time.

A polynomial oracle-time algorithm for convex integer minimization

We consider the following Colourful generalization of Linear Programming: given sets of points S1,..., Sk ⊂ Rd, referred to as colours, and a point b ∈ Rd, decide whether there is a colourfulT = {s1,..., sk} such that b ∈ convT, and if there is one, find it. Linear Programming is obtained by taking k = d + 1 and S1 =... = Sd+1. If k = d + 1 and b ∈ ∩i=1d+1 convSi then a solution always exists: we describe an efficient iterative approximation algorithm for this problem, that finds a colourful T whose convex hull contains a point e-close to b, and analyze its real arithmetic and Turing time complexities. In contrast, we show that Colourful Linear Programming is strongly NP-complete. We consider a class of linear algebraic relatives of Colourful Linear Programming, and give a computational complexity classification of the related decision and counting problems that arise. We also introduce and discuss the complexity of a hierarchy of w1, w2-Matroid-Basis-Nonbasis problems, and give an application of Colourful Linear Programming to the algorithmic problems of Tverbergs theorem in combinatorial geometry.

The Vector Partition Problem for Convex Objective Functions

We show that any rational convex polytope is polynomial-time representable as a 3-way line-sum transportation polytope of “slim”

Graphs of transportation polytopes

(r,c,3)

Nonlinear Matroid Optimization and Experimental Design

format. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. We provide a polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bitransportation programs. Our construction resolves several standing problems on