🤖 AI Summary
This paper studies the parameterized Vertex Cover problem, aiming to break the long-standing O*(1.2738^k) time-complexity lower bound established by Chen, Kanj, and Xia (2010). We introduce the first potential function jointly tracking both the solution size k and the optimal value λ of the LP relaxation. Based on this, we design novel branching rules to overcome local obstructions in the search tree. Our algorithm integrates LP-based preprocessing, structural analysis of maximum independent sets, and over-approximation techniques for vertex cover. Through meticulous potential-function analysis, we achieve tight control over the branching process in the branch-and-bound framework. The resulting deterministic algorithm runs in O*(1.25284^k) time—currently the fastest known parameterized algorithm for Vertex Cover—and significantly advances the theoretical time bound for this fundamental problem.
📝 Abstract
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.