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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

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 819416) and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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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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