Recognizing Unit Multiple Intervals Is Hard

Authors Virginia Ardévol Martínez , Romeo Rizzi , Florian Sikora , Stéphane Vialette

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Author Details

Virginia Ardévol Martínez
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Romeo Rizzi
  • Department of Computer Science, University of Verona, Italy
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France
Stéphane Vialette
  • LIGM, CNRS, Univ Gustave Eiffel, F77454 Marne-la-Vallée, France


Part of this work was conducted when RR was an invited professor at Université Paris-Dauphine.

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Virginia Ardévol Martínez, Romeo Rizzi, Florian Sikora, and Stéphane Vialette. Recognizing Unit Multiple Intervals Is Hard. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Multiple interval graphs are a well-known generalization of interval graphs introduced in the 1970s to deal with situations arising naturally in scheduling and allocation. A d-interval is the union of d intervals on the real line, and a graph is a d-interval graph if it is the intersection graph of d-intervals. In particular, it is a unit d-interval graph if it admits a d-interval representation where every interval has unit length. Whereas it has been known for a long time that recognizing 2-interval graphs and other related classes such as 2-track interval graphs is NP-complete, the complexity of recognizing unit 2-interval graphs remains open. Here, we settle this question by proving that the recognition of unit 2-interval graphs is also NP-complete. Our proof technique uses a completely different approach from the other hardness results of recognizing related classes. Furthermore, we extend the result for unit d-interval graphs for any d ⩾ 2, which does not follow directly in graph recognition problems -as an example, it took almost 20 years to close the gap between d = 2 and d > 2 for the recognition of d-track interval graphs. Our result has several implications, including that recognizing (x, …, x) d-interval graphs and depth r unit 2-interval graphs is NP-complete for every x ⩾ 11 and every r ⩾ 4.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph theory
  • Interval graphs
  • unit multiple interval graphs
  • recognition
  • NP-hardness


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