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Bandwidth Parameterized by Cluster Vertex Deletion Number

Authors Tatsuya Gima , Eun Jung Kim , Noleen Köhler , Nikolaos Melissinos , Manolis Vasilakis

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Tatsuya Gima
  • JSPS Research Fellow, Nagoya University, Japan
Eun Jung Kim
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Noleen Köhler
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France
Nikolaos Melissinos
  • Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Czech Republic
Manolis Vasilakis
  • Université Paris-Dauphine, PSL University, CNRS UMR7243, LAMSADE, Paris, France

Funding

Our research visit to Nagoya University, Japan was funded by PRC CNRS JSPS project PARAGA (Parameterized Approximation Graph Algorithms).
  • Gima, Tatsuya: Partially supported by JSPS KAKENHI Grant Number JP23KJ1066.
  • Kim, Eun Jung: Supported by ANR project ANR-18-CE40-0025-01 (ASSK).
  • Köhler, Noleen: Supported by ANR project ANR-18-CE40-0025-01 (ASSK).
  • Melissinos, Nikolaos: Supported by the CTU Global postdoc fellowship program.
  • Vasilakis, Manolis: Partially supported by ANR project ANR-21-CE48-0022 (S-EX-AP-PE-AL).

Acknowledgements

We would like to thank Virginia Ardévol Martínez and Yota Otachi for interesting discussions at the preliminary stages of this work.

Cite AsGet BibTex

Tatsuya Gima, Eun Jung Kim, Noleen Köhler, Nikolaos Melissinos, and Manolis Vasilakis. Bandwidth Parameterized by Cluster Vertex Deletion Number. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.IPEC.2023.21

Abstract

Given a graph G and an integer b, Bandwidth asks whether there exists a bijection π from V(G) to {1, …, |V(G)|} such that max_{{u, v} ∈ E(G)} | π(u) - π(v) | ≤ b. This is a classical NP-complete problem, known to remain NP-complete even on very restricted classes of graphs, such as trees of maximum degree 3 and caterpillars of hair length 3. In the realm of parameterized complexity, these results imply that the problem remains NP-hard on graphs of bounded pathwidth, while it is additionally known to be W[1]-hard when parameterized by the treedepth of the input graph. In contrast, the problem does become FPT when parameterized by the vertex cover number of the input graph. In this paper, we make progress towards the parameterized (in)tractability of Bandwidth. We first show that it is FPT when parameterized by the cluster vertex deletion number cvd plus the clique number ω of the input graph, thus generalizing the previously mentioned result for vertex cover. On the other hand, we show that Bandwidth is W[1]-hard when parameterized only by cvd. Our results generalize some of the previous results and narrow some of the complexity gaps.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Bandwidth
  • Clique number
  • Cluster vertex deletion number
  • Parameterized complexity

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