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Mim-Width Is paraNP-Complete

Authors Benjamin Bergougnoux , Édouard Bonnet , Julien Duron

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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Fixed parameter tractability
Keywords
  • Mim-width
  • lower bounds
  • parameterized complexity
  • ordered graphs

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Abstract

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant. A key intermediate problem that we introduce and show NP-complete, Linear Degree Balancing, inputs an edge-weighted graph G and an integer τ, and asks whether V(G) can be linearly ordered such that every vertex of G has weighted backward and forward degrees at most τ.

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Benjamin Bergougnoux, Édouard Bonnet, and Julien Duron. Mim-Width Is paraNP-Complete. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.25

Author Details

Benjamin Bergougnoux
  • LIS, Aix-Marseille Université, France
Édouard Bonnet
  • CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France
Julien Duron
  • ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR 5668, Lyon, France

Funding

ÉB and JD were supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014-01) and DIGRAPHS (ANR-19-CE48-0013-01).

References

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