Recognizing unit multiple interval graphs is hard

• Published in: Discrete Applied Mathematics Vol. 360; pp. 258 - 274
• Main Authors: Ardévol Martínez, Virginia, Rizzi, Romeo, Sikora, Florian, Vialette, Stéphane
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.01.2025; Elsevier
• Subjects: [formula omitted]-hardness; Computer Science; Interval graphs; Recognition; Unit multiple interval graphs; Interval graphs; NP-hardness; Unit multiple interval graphs; Recognition
• DOI: 10.1016/j.dam.2024.09.011
• ISSN: 0166-218X

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 disjoint 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 for every d≥2, recognizing (x,…,x)d-interval graphs and depth r unit d-interval graphs is NP-complete for every x≥11 and every r≥4.