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FPT and Kernelization Algorithms for the Induced Tree Problem

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Algorithms and Complexity (CIAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12701))

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Abstract

The three-in-a-tree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has become a critical subroutine in many algorithms for detecting induced subgraphs, such as beetles, pyramids, thetas, and even and odd-holes. In 2007, Derhy and Picouleau [Discrete Applied Mathematics, 2009] considered the natural generalization to k mandatory vertices with k being part of the input, and showed that it is \(\mathsf {NP}\)-\(\mathsf {complete}\); they named this problem Induced Tree, and asked what is the complexity of four-in-a-tree. Motivated by this question and the relevance of the original problem, we study the parameterized complexity of Induced Tree. We begin by showing that the problem is \(\mathsf {W}\)[1]-\(\mathsf {hard}\) when jointly parameterized by the size of the solution and minimum clique cover and, under the Exponential Time Hypothesis, does not admit an \(n^{o(k)}\) time algorithm. Afterwards, we use Courcelle’s Theorem to prove tractability under cliquewidth, which prompts our investigation into which parameterizations admit single exponential algorithms; we show that such algorithms exist for the unrelated parameterizations treewidth, distance to cluster, and vertex deletion distance to co-cluster. In terms of kernelization, we present a linear kernel under feedback edge set, and show that no polynomial kernel exists under vertex cover nor distance to clique unless \({\mathsf {NP}}\subseteq {\mathsf {coNP}}/{\mathsf {poly}}\). Along with other remarks and previous work, our tractability and kernelization results cover many of the most commonly employed parameters in the graph parameter hierarchy.

Work partially supported by CAPES, CNPq, and FAPEMIG. Full version permanently available at https://arxiv.org/abs/2007.04468.

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Notes

  1. 1.

    The width of a bipartition (A, B) of V(G) is the number of edges between the parts. The bisection width of G is equal to the minimum width among all bipartitions of V(G) where \(|A| \le |B| \le |A|+1\).

  2. 2.

    A layout of G is a bijection \(f : V(G) \mapsto [n]\); the width of f is given by \(\max _{uv \in E(G)}|f(u) - f(v)|\). The bandwidth of G is equal to the width of a minimum-width layout.

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Authors and Affiliations

  1. Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Guilherme Castro Mendes Gomes & Vinicius F. dos Santos

  2. Departamento de Informática, Universidade Federal do Paraná, Curitiba, Brazil

    Murilo V. G. da Silva

  3. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

  4. Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil

    Jayme L. Szwarcfiter

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  1. Guilherme Castro Mendes Gomes
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  2. Vinicius F. dos Santos
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  3. Murilo V. G. da Silva
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  4. Jayme L. Szwarcfiter
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Correspondence to Guilherme Castro Mendes Gomes .

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  1. Sapienza University of Rome, Rome, Italy

    Tiziana Calamoneri

  2. Sapienza University of Rome, Rome, Italy

    Federico Corò

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Gomes, G.C.M., dos Santos, V.F., da Silva, M.V.G., Szwarcfiter, J.L. (2021). FPT and Kernelization Algorithms for the Induced Tree Problem. In: Calamoneri, T., Corò, F. (eds) Algorithms and Complexity. CIAC 2021. Lecture Notes in Computer Science(), vol 12701. Springer, Cham. https://doi.org/10.1007/978-3-030-75242-2_11

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