On the Complexity of Quantified Integer Programming

Quantified integer programming is the problem of deciding assertions of the form Q_k x_k ... forall x_2 exists x_1 : A * x >= c where vectors of variables x_k,..,x_1 form the vector x, all variables are interpreted over N (alternatively, over Z), and A and c are a matrix and vector over Z of appropriate sizes. We show in this paper that quantified integer programming with alternation depth k is complete for the kth level of the polynomial hierarchy

N-fold integer programming in cubic time

N-fold integer programming is a fundamental problem with a variety of natural applications in operations research and statistics. Moreover, it is universal and provides a new, variable-dimension, parametrization of all of integer programming. The fastest algorithm for $n$-fold integer programming predating the present article runs in time $O(n^{g(A)}L)$ with $L$ the binary length of the numerical part of the input and $g(A)$ the so-called Graver complexity of the bimatrix $A$ defining the system. In this article we provide a drastic improvement and establish an algorithm which runs in time $O(n^3 L)$ having cubic dependency on $n$ regardless of the bimatrix $A$. Our algorithm can be extended to separable convex piecewise affine objectives as well, and also to systems defined by bimatrices with variable entries. Moreover, it can be used to define a hierarchy of approximations for any integer programming problem

A note on peer-to-peer satellite refueling strategies.

We revisit the peer-to-peer refueling problem, in which the maneuvering satellites are allowed to interchange their orbital positions. We show that the problem is computationally hard, by reducing it to a special case of the three-index assignment problem. On the positive side, we show that the size of instances from practice is such that a state-of-the-art integer programming solver is able to find optimal solutions in little computing time.Three-index assignment problem; Complexity; Integer programming;

A branch-and-price algorithm for a hierarchical crew scheduling problem.

We describe a real-life problem arising at a crane rental company. This problem is a generalization of the basic crew scheduling problem given in Mingozzi et al. and Beasley and Cao. We formulate the problem as an integer programming problem and establish ties with the integer multicommodity flow problem and the hierarchical interval scheduling problem. After establishing the complexity of the problem we propose a branch-and-price algorithm to solve it. We test this algorithm on a limited number of real-life instances.Scheduling;