Rational Jacobi Rotations and the Complexity of Approximating Mixed Integer Quadratic Programming

2026-07-31

AI Summary

This work establishes the first characterization of the ε-approximation complexity for mixed-integer quadratic programming (MIQP) under the Turing machine model, assuming a fixed number of integer variables and a fixed number of negative eigenvalues in the Hessian matrix of the objective function. The authors propose a polynomial-time algorithm based on rational Jacobi rotations that constructs a rational basis transformation preserving the inertia of the Hessian, thereby simultaneously diagonalizing the objective function and exactly mapping the associated ellipsoid to a sphere. Relying solely on rational arithmetic, this approach overcomes the traditional reliance on irrational transformations. Consequently, it achieves polynomial-time ε-approximate solutions in terms of both the input size and 1/ε, with the result also extending to the purely continuous case.

Abstract

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.

Problem

• Mixed Integer Quadratic Programming
• Approximation Complexity
• Rational Jacobi Rotations
• Hessian Inertia
• Polynomial-Time Approximation

Innovation

• rational Jacobi rotations
• simultaneous diagonalization
• mixed integer quadratic programming
• polynomial-time approximation
• Hessian inertia