A faster algorithm for Vertex Cover parameterized by solution size

Abstract

We describe a new algorithm for vertex cover with runtime O * (1.25400 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 λ 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 µ = k − λ are decreased at each step. There can be local obstructions in the graph that prevent µ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

Parameterized Algorithms for Colored Clustering

“…As shown by Cai and Leung (2018), COLORED CLUSTER-ING can be solved in time O * (1.2783 r ), where r := |E| − k is the number of unstable edges, by reduction to VERTEX COVER. The recently proposed O * (1.2530 k ) time algorithm for VERTEX COVER (Harris and Narayanaswamy 2022) can be used to improve this running time. There is a line of research that seeks to improve on such algorithms by introducing smaller parameters that take a lower bound on the parameter and then use the difference between the traditional parameter and the lower bound as a smaller parameter.…”