XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

Authors Hans L. Bodlaender , Carla Groenland , Hugo Jacob , Lars Jaffke , Paloma T. Lima

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Author Details

Hans L. Bodlaender
  • Utrecht University, The Netherlands
Carla Groenland
  • Utrecht University, The Netherlands
Hugo Jacob
  • ENS Paris-Saclay, France
Lars Jaffke
  • University of Bergen, Norway
Paloma T. Lima
  • IT University of Copenhagen, Denmark

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Hans L. Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, and Paloma T. Lima. XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Parameterized complexity and exact algorithms
  • parameterized complexity
  • XNLP
  • linear clique-width
  • W-hierarchy
  • pathwidth
  • linear mim-width
  • bandwidth


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