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

Hans Bodlaender, Carla Groenland, Hugo Jacob, Lars Jaffke, Paloma Thomé de Lima

Research output: Conference Article in Proceeding or Book/Report chapterArticle in proceedingsResearchpeer-review


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.
Original languageEnglish
Title of host publication17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Publication date2022
Publication statusPublished - 2022


  • XNLP
  • linear width measures
  • parameterized complexity
  • hardness proofs
  • XP space requirement


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