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 of our main result we show that Optimal Linear Arrangement is fixed parameter tractable parameterized by the size of the smallest vertex cover of the input graph. This resolves an open problem from the recent monograph by Downey and Fellows.