发表机构
Georgia Institute of Technology（佐治亚理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出随机多项式时间算法，用 $O(n^{4/23})$ 种颜色为 3-可着色图着色，通过高斯覆盖递归和方差界改进现有界。
AI 中文摘要
我们给出一个随机多项式时间算法，可为任意承诺为 3-可着色的 $n$ 个顶点的图用 $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ 种颜色着色，改进了近期由 Bansal、Huang 和 Lee 以及 Narang 和 Tang 得到的 $O(n^{0.19539})$ 的界，后者对每个固定的 $\smash{\epsilon>0}$ 获得了 $O(n^{(13-\sqrt{97})/18+\epsilon})=O(n^{0.17506\dots + \epsilon})$ 种颜色的结果。为证明我们的结果，我们从固定层次的半定松弛出发，其中我们在高斯覆盖上使用有限深度的递归。固定一个根顶点后，我们根据与根向量的相关性将顶点分组。这里，每一步通过一条边扩展一个方向覆盖并将其转移到一个后继组。我们的关键分析成分是高斯最大值的一个方差界：对于 $m\geq 2$ 个系数范数至多为 $r$、均值为 $\mu$、方差为 $v$ 的中心线性形式的最大值，我们利用 Chen 的高斯凸性定理证明 $v\leq r^2-\mu^2/(2\log m)$。结合一个方差尺度的下尾估计，这控制了每次扩展时的阈值损失，这表明根条件向量着色要么能从一组中提取一个大的独立集，要么限制其大小，从而在常数步后迫使矛盾。所得的稀疏情形保证与 Kawarabayashi、Thorup 和 Yoneda 的稠密进展界相结合，且递归的数值不等式通过有理区间算术得到验证。
英文摘要
We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+ε})=O(n^{0.17506\dots + ε})$ colors for every fixed $\smash{ε>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\leq r^2-μ^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.
Comments29 pages, 1 figure
