We describe a new algorithm for vertex cover with runtime $O^*(1.25288^k)$, 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:我们描述一个新的顶点覆盖算法, 其运行时间为$[( 1.25288]k]$, 美元是理想溶液的大小, 美元隐藏输入大小的多元系数。 这比先前运行时间( 1. 2738]k)$( 1.2738美元) 有所改进, 因为Chen, Kanj, & Xia (2010年) 站立了十多年。 我们算法的关键是使用一个潜在的函数, 该函数同时跟踪$( k) 以及顶点覆盖LP 放松的最佳值 $\lambda美元 。 这种方法还允许我们使用前一个在约束度图形和高担保度 Vertex 覆盖中为最大独立设置的算法 。 算法的主要步骤是在高度脊椎上分支, 同时确保每步都减少 $( k) 和 $\ mu = k -\ lambda$。 。 图表中可能存在防止 $\ mu$( mu$) 减少此过程的局部障碍 ; 我们开发了一些处理这些情况的新分支步骤 。