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：本文描述了一种新的顶点覆盖算法，其运行时间为$O^*(1.25284^k)$，其中$k$是期望解的大小，$O^*$隐藏了输入规模的多项式因子。该结果改进了 Chen、Kanj 和 Xia（2010）提出的$O^*(1.2738^k)$运行时间，该纪录已保持十余年。我们算法的关键在于使用一个势函数同时追踪$k$和顶点覆盖线性规划松弛的最优值$\lambda$。这一方法还使我们得以利用有界度图中最大独立集问题的先有算法以及超保证顶点覆盖算法。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$\mu = k - \lambda$均减小。图中可能存在局部障碍阻止$\mu$在此过程中减小；我们开发了若干新颖的分支步骤来处理这些情况。