Improved FPT Algorithms for Deletion to Forest-Like Structures

Authors Kishen N. Gowda, Aditya Lonkar, Fahad Panolan, Vraj Patel, Saket Saurabh



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Author Details

Kishen N. Gowda
  • IIT Gandhinagar, India
Aditya Lonkar
  • IIT Madras, India
Fahad Panolan
  • Department of Computer Science and Engineering, IIT Hyderabad, India
Vraj Patel
  • IIT Gandhinagar, India
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India

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Kishen N. Gowda, Aditya Lonkar, Fahad Panolan, Vraj Patel, and Saket Saurabh. Improved FPT Algorithms for Deletion to Forest-Like Structures. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.34

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). 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 [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^𝓁, 2.7^k ⋅ 36.61^𝓁, 3^k ⋅ 1.78^𝓁}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • Independent Feedback Vertex Set
  • PseudoForest
  • Almost Forest
  • Cut and Count
  • Treewidth

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