  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.
