论文标题
精确的半决赛编程范围用于包装问题
Exact semidefinite programming bounds for packing problems
论文作者
论文摘要
在本文中，我们给出了一种算法，以将半决赛编程求解器的浮点输出绕到理由或二次延伸的解决方案。我们将其应用于包装问题的尖锐界限，我们使用这些锋利的边界来证明某些最佳包装配置是独特的。特别是，我们表明配置来自$ \ Mathsf {e} _8 $ root lattice是一个独特的最佳代码\ vartheta =（2 \ sqrt {2} -1）/7 $，通过舍入到$ \ mathbb q [\ sqrt {2}] $，是清晰的。我们还使用我们的机械计算可以包装成更大球体的球体数量上的锋利上限。
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $π/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.
