Optimization over l×m× n integer threeway tables is NP-hard already for fixed l = 3, but solvable in polynomial time with both l,m fixed. Here we consider huge tables, where the variable dimension n is encoded in binary. Combining recent results on Graver bases and recent results on integer cones, we show how to handle such problems in polynomial time. We also show that a harder variant of the ...

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 so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several appl...

V-variable fractals, where V is a positive integer, are intuitively fractals with at most V different “forms” or “shapes” at all levels of magnification. In this paper we describe how V-variable fractals can be used for the purpose of image compression.

Branching is an important component of the branch-and-cut algorithm for solving mixed integer linear programs. Most solvers branch by imposing a disjunction of the form“xi ≤ k ∨ xi ≥ k + 1” for some integer k and some integer-constrained variable xi. A generalization of this branching scheme is to branch by imposing a more general disjunction of the form “πx ≤ π0 ∨ πx ≥ π0 + 1.” In this paper, ...

We consider N-fold 4-block decomposable integer programs, which simultaneously generalize N-fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Köppe, R. Weismantel, A polynomial-time algorithm for optimizing over N-fold 4block decomposable integer programs, Proc. IPCO 2010, Lecture Notes in Computer Science, vol. 6080, Springer, 2...

How large is an integer that a computer can take? Certainly it depends on what software that you are going to use. For instance, the largest unsigned long integer in C++ is 1 2 − (see [1]), while it is 1 2 − in C# (see [2]). If a variable in C++ is declared to be of unsigned long integer type, it has the maximum value of 4,294,967,295. It will result in an integer overflow when a program in C++...