Improved FPT Algorithms for Deletion to Forest-Like Structures

Abstract

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 , 2.7 k • 36.61 , 3 k • 1.78 }).