改进的基于递归高斯证书的 3-可着色图 SDP 着色
Improved SDP Coloring of 3-Colorable Graphs from Recursive Gaussian Certificates
- Georgia Institute of Technology, School of Computer Science（佐治亚理工学院计算机学院）
- Princeton University, Operations Research and Financial Engineering（普林斯顿大学运筹与金融工程学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种随机多项式时间算法，利用递归高斯证书分析高层邻域，将3-可着色图的着色数改进至O(n^{0.17506+ε})，优于此前最优界。
AI中文摘要：
我们给出一个随机多项式时间算法，对于每个固定的 $\varepsilon > 0$，用 $O\bigl(n^{(13-\sqrt{97})/18+\varepsilon}\bigr) \approx O\bigl(n^{0.17506+\varepsilon}\bigr)$ 种颜色为每个 $3$-可着色的 $n$ 顶点图着色，改进了 Bansal、Huang 和 Lee 之前的最佳界 $O(n^{0.19539})$。我们的改进来自通过高斯 SDP 舍入中失败的递归描述来分析更高层的邻域。如果舍入返回的独立集太小，它会在非空诱导子图的每个顶点产生局部高斯证书。我们沿着游走将这些证书传播到更高层的邻域，通过定义递归证书结构并证明一个加强的覆盖-组合引理（该引理细化了 Arora、Chlamtáč 和 Charikar 的引理）。然后我们构造一个有界势函数，它在每一步传播中增加一个固定的正量，从而产生矛盾。因此，舍入必须产生一个足够大的独立集。
英文摘要：
We give a randomized polynomial-time algorithm that, for every fixed $\varepsilon > 0$, colors every $3$-colorable $n$-vertex graph using $O\bigl(n^{(13-\sqrt{97})/18+\varepsilon}\bigr) \approx O\bigl(n^{0.17506+\varepsilon}\bigr)$ colors, improving upon the previous best bound of $O(n^{0.19539})$ from Bansal, Huang, and Lee. Our improvement comes from analyzing higher-level neighborhoods through a recursive description of failure in Gaussian SDP rounding. If the rounding returns too small an independent set, it produces local Gaussian certificates at every vertex of a nonempty induced subgraph. We propagate these certificates along walks to higher-level neighborhoods by defining a recursive certificate structure and proving a strengthened cover-composition lemma, which refines the one of Arora, Chlamt{á}{č}, and Charikar. We then construct a bounded potential function that increases by a fixed positive amount at every propagation step, yielding a contradiction. Consequently, the rounding must produce a sufficiently large independent set.
