An improved exact algorithm for undirected feedback vertex set - Xiao, Mingyu; Nagamochi, Hiroshi (Journal of combinatorial optimization, 2015)
A feedback vertex set in an undirected graph is a subset of vertices removal of which leaves a graph with no cycles. Razgon (in: Proceedings of the 10th Scandinavian workshop on algorithm theory (SWAT 2006), pp. 160–171, 2006 ) gave a 1 . 8899 n n O ( 1 ) -time algorithm for finding a minimum feedback vertex set in an n -vertex undirected graph, which is the first exact algorithm for the problem that breaks the trivial barrier of 2 n . Later, Fomin et al. (Algorithmica 52:293–307, 2008 ) improved the result to 1 . 7548 n n O ( 1 ) . In this paper, we further improve the result to 1 . 7266 n n O ( 1 ) . Our algorithm is analyzed by the measure-and-conquer method. We get the improvement by designing new reductions based on biconnectivity of instances and introducing a new measure scheme on the structure of reduced graphs.