Bansal--Huang--Lee 着色框架中更精确的高斯覆盖

中文摘要 AI 辅助

Bansal、Huang 和 Lee 最近给出了一种多项式时间算法，能够用 $O(n^{0.19539})$ 种颜色对任意 $n$ 个顶点的 $3$-可着色图进行着色。我们将这一界改进为 $O(n^{0.17794})$ 种颜色。改进来源于一个更精确的高斯覆盖组合规则。我们通过比较高斯向量在两个阈值下错过一个多面体的概率，从 Ehrhard 不等式证明了该规则。在较远的阈值处，联合界记录了定义该多面体的半空间数量，这一信息减少了组合中的损失。然后我们将 Bansal-Huang-Lee 论证中连续的邻域步骤组织成单一递归。较小的损失使得递归比之前多运行一步，而在该步骤中所需的不同顶点数将超过图中所包含的顶点数，从而排除了剩余情况。将由此得到的有限度保证与 Kawarabayashi、Thorup 和 Yoneda 的稠密图算法结合，得到一种随机多项式时间算法，能够用 $O(n^{0.17794})$ 种颜色对任意 $n$ 个顶点的 $3$-可着色图进行着色。

英文摘要

Bansal, Huang, and Lee recently gave a polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.19539})$ colors. We improve their bound to $O(n^{0.17794})$ colors. The improvement comes from a sharper rule for combining Gaussian covers. We prove the rule from Ehrhard's inequality by comparing the probability that a Gaussian vector misses a polyhedron at two thresholds. At the farther threshold a union bound records how many halfspaces define the polyhedron, and this information reduces the loss in the combination. We then organize the successive neighborhood steps of the Bansal-Huang-Lee argument into a single recursion. The smaller loss lets the recursion run one step farther than before, and at that step it would require more distinct vertices than the graph contains. This rules out the remaining case. Combining the resulting bounded-degree guarantee with the dense-graph algorithm of Kawarabayashi, Thorup, and Yoneda gives a randomized polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.17794})$ colors.

发表机构

• Northwestern University（西北大学）
• TTIC（泰格特计算机科学研究所）

机构由 AI 辅助整理，请以论文原文为准。