Recognizing unit multiple interval graphs is hard

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 $\mathrm{NP}$-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 $\mathrm{NP}$-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 $d\ge 2$, 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 $d\ge 2$, recognizing $(x,\dots ,x)$ $d$-interval graphs and depth $r$ unit $d$-interval graphs is $\mathrm{NP}$-complete for every $x\ge 11$ and every $r\ge 4$.