We present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective function are fixed. Unless P=NP, both restrictions are necessary, so this completes the characterization of the complexity of approximating MIQP in terms of the number of integer variables and the inertia of the Hessian; the result is new already in the purely continuous case. The main ingredient is a polynomial-time simultaneous diagonalization algorithm: it computes a rational change of basis that maps a given ellipsoid, presented in factored form, exactly to a ball, while making the objective function separable up to an arbitrarily small perturbation and preserving the inertia of its Hessian. The classical construction, in which the objective function is made exactly separable, requires a change of basis that is in general irrational, and cannot be carried out on a Turing machine; ours rests instead on rational Jacobi rotations, which we believe to be of independent interest.

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