Integer programming in parameterized complexity: Five miniatures

Parameterized Integer Quadratic Programming: Variables and Coefficients

In the Integer Quadratic Programming problem input is an n× n integer matrix Q, an m × n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x ∈ Z minimizing xQx, subject to Ax ≤ b. We give a fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by n+α. Here α is the largest absolute value of an entry of Q and A. As an application o...

Parameterized Resiliency Problems via Integer Linear Programming

We introduce a framework in parameterized algorithms whose purpose is to solve resiliency versions of decision problems. In resiliency problems, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for I...

Complexity of Primal Methods in Integer Programming

One common approach to solve optimization problems is the primal method. One starts with a feasible point and then successively produces new feasible solutions with better objective function values until an optimal solution is reached. From an abstract point of view, an augmentation problem is solved in each iteration, i.e., given a feasible point find an augmenting vector, if one exists. The d...

Integer Programming Models and Parameterized Algorithms for Controlling Palletizers

We study the combinatorial FIFO Stack-Up problem, where bins have to be stacked-up from conveyor belts onto pallets. This is done by palletizers or palletizing robots. Given k sequences of labeled bins and a positive integer p, the goal is to stack-up the bins by iteratively removing the first bin of one of the k sequences and put it onto a pallet located at one of p stack-up places. Each of th...

Quadratic Integer Programming: Complexity and Equivalent Forms