有界整数二次规划：通过平行六面体覆盖与离散凸优化
Bounded Integer Quadratic Programming through Parallelepiped Covers and Discrete Convic Optimization
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中文总结 AI 辅助
提出精确算法，通过细化平行六面体覆盖与离散凸优化，在多项式时间内求解有界有理多面体上的任意有理二次整数规划，并扩展至无界情形。
中文摘要 AI 辅助
我们给出一个精确算法，用于在任意有界有理多面体 $\{x: Ax \le b\}$ 的整数点上最小化任意有理二次多项式 $x^T Q x + c^T x + \gamma$。对于 $n$ 个变量和 $m$ 个不等式，运行时间为 $$2^{O(n \log(n+1))} (m+1)^{O(n)} \beta^{O(n)} (1+L)^{O(1)},$$ 其中 $\beta$ 是 $A$ 或 $Q$ 的条目最大二进制编码长度加一，$L$ 是完整输入编码长度。关于 $L$ 的多项式次数是绝对的；特别地，$b$、$c$ 和 $\gamma$ 的编码仅进入这个固定次数因子。我们的算法将 Goemans 和 Rothvoss 的整数平行六面体覆盖细化为具有受控位移集的单元。在每个单元上，要么负二次值的整数位移排除每个点作为全局最小化器，要么二次恒等式为每个整数目标子水平集的凸包提供分离预言。然后 Hildebrand 和 Goess 的整数查询可行性定理找到单元最小值。通过缩放格网的细化解决位移搜索，定位论证将约束矩阵和二次部分的编码依赖与其余数据的编码依赖分离。将有界算法与 Ari 和 Hildebrand 的无界性测试相结合，将方法扩展到任意有理多面体。
英文摘要
We give an exact algorithm for minimizing an arbitrary rational quadratic polynomial $x^T Q x + c^T x + γ$ over the integer points of a bounded rational polyhedron $\{x : Ax \le b\}$. For $n$ variables and $m$ inequalities, the running time is $$2^{O(n \log(n+1))} (m+1)^{O(n)} β^{O(n)} (1+L)^{O(1)},$$ where $β$ is one plus the maximum binary encoding length of an entry of $A$ or $Q$, and $L$ is the full input encoding length. The polynomial degree in $L$ is absolute; in particular, the encoding of $b$, $c$, and $γ$ enters only this fixed-degree factor. Our algorithm refines the integral parallelepiped cover of Goemans and Rothvoss into cells with controlled displacement sets. On each cell, either an integer displacement of negative quadratic value rules out every point as a global minimizer, or the quadratic identity supplies a separation oracle for the convex hull of each integer objective sublevel set. The integer-query feasibility theorem of Hildebrand and Goess then finds the cell minimum. A refinement through scaled lattices solves the displacement search, and a localization argument separates the encoding dependence of the constraint matrix and quadratic part from that of the remaining data. Combining the bounded algorithm with the unboundedness test of Ari and Hildebrand extends the method to arbitrary rational polyhedra.
