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 \(S\subseteq V(G)\) 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 \({\mathcal {O}}^{\star }(2.7^k)^{*}\). 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 \(k,\ell \in {{\mathbb {N}}}\), and the objective is to test whether there exists a vertex subset S of size at most k, such that \(G-S\) is \(\ell \) 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.

1. 1.

   Independent Feedback Vertex Set can be solved in time \({\mathcal {O}}^{\star }(2.7^k)\).

2. 2.

   Pseudo Forest Deletion can be solved in time \({\mathcal {O}}^{\star }(2.85^k)\).

3. 3.

   Almost Forest Deletion can be solved in time \({\mathcal {O}}^{\star }(\min \{2.85^k \cdot 8.54^\ell ,2.7^k \cdot 36.61^\ell ,3^k \cdot 1.78^\ell \})\).