Improved FPT Algorithms for Deletion to Forest-Like Structures

The FEEDBACK VERTEX SET problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset S ⊆ V(G) of size at most k such that G − S is a forest. After a long line of improvement, recently, Li and Nederlof [TALG, 2022] designed a randomized algorithm for the problem running in time (O*2.7ᵏ). In the Parameterized Complexity literature, several problems around FEEDBACK VERTEX SET have been studied. Some of these include INDEPENDENT FEEDBACK VERTEX SET (where the set S should be an independent set in G), ALMOST FOREST DELETION and PSEUDOFOREST DELETION. In PSEUDOFOREST DELETION, each connected component in G − S has at most one cycle in it. However, in ALMOST FOREST DELETION, the input is a graph G and non-negative integers k, ℓ ∈ N, and the objective is to test whether there exists a vertex subset S of size at most k, such that G − S is ℓ edges away from a forest. In this paper, using the methodology of Li and Nederlof [TALG, 2022], we obtain the current fastest algorithms for all these problems. In particular we obtain the following randomized algorithms.