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 [SODA, 2020] designed a randomized algorithm for the problem running in time O^?(2.7^k)¹. 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, ` ∈ N, and the objective is to test whether there exists a vertex subset S of size at most k, such that G − S is ` edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms. 1. Independent Feedback Vertex Set can be solved in time O^?(2.7^k). 2. Pseudo Forest Deletion can be solved in time O^?(2.85^k). 3. Almost Forest Deletion can be solved in O^?(min{2.85^k · 8.54^{`</sup>, 2.7<sup>k</sup> · 36.61<sup>`}, 3^k · 1.78^`}).