We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $\lambda$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $\mu = k - \lambda$ are decreased at each step. There can be local obstructions in the graph that prevent $\mu$ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

翻译：我们描述了一种新的顶点覆盖算法，运行时间为 $O^*(1.25284^k)$，其中 $k$ 是所需解的大小，$O^*$ 隐藏了输入规模中的多项式因子。这改进了 Chen、Kanj 和 Xia（2010）提出的之前运行时间为 $O^*(1.2738^k)$ 的算法，该结果已保持十余年。我们算法的关键在于使用了一个势函数，该函数同时追踪 $k$ 以及顶点覆盖线性规划松弛的最优值 $\lambda$。这种方法还使我们能够利用有界度图中最大独立集和超保证顶点覆盖的先前算法。算法的主要步骤是对高度数顶点进行分支，同时确保每一步中 $k$ 和 $\mu = k - \lambda$ 都减小。在此过程中，图中可能存在阻止 $\mu$ 减小的局部障碍；我们开发了多种新颖的分支步骤来处理这些情况。