有理雅可比旋转与混合整数二次规划近似的复杂性
Rational Jacobi Rotations and the Complexity of Approximating Mixed Integer Quadratic Programming
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中文总结 AI 辅助
该研究提出基于有理雅可比旋转的多项式时间算法，可在整数变量与目标函数海森负特征值数量固定时求混合整数二次规划的ε-近似解，完整刻画其近似复杂性，纯连续情形下结果为新。
中文摘要 AI 辅助
我们提出一种算法，该算法可求混合整数二次规划（MIQP）问题的ε-近似解，且在图灵机上的运行时间为实例规模与1/ε的多项式时间，前提是整数变量的数量及目标函数海森矩阵的负特征值数量固定。除非P=NP，否则这两项限制均为必要条件，因此该成果从整数变量数量与海森矩阵惯性的角度完整刻画了MIQP近似的复杂性；该结果在纯连续情形下也是全新的。核心要素是一种多项式时间同时对角化算法：它计算一个有理基变换，该变换可将以因式形式给出的给定椭球精确映射为球，同时使目标函数在任意小扰动下可分，并保留其海森矩阵的惯性。经典构造中，目标函数被精确可分所需的基变换通常是无理的，无法在图灵机上实现；而我们的方法则基于有理雅可比旋转，我们认为该旋转本身具有独立研究价值。
英文摘要
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.
