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

翻译：多重区间图是区间图的一个著名推广，于20世纪70年代提出，用于处理调度和分配中自然出现的情形。一个$d$-区间是实直线上$d$个区间的并集，如果一个图是$d$-区间的交图，则称其为$d$-区间图。特别地，若它允许一个每个区间长度均为单位的$d$-区间表示，则称为单位$d$-区间图。尽管长期以来已知识别2-区间图及其他相关类别（如2-轨道区间图）是NP完全的，但识别单位2-区间图的复杂性仍悬而未决。在此，我们通过证明识别单位2-区间图也是NP完全的，解决了这一问题。我们的证明技术采用了与识别相关类别的其他难度结果完全不同的方法。此外，我们将结果推广到任意$d\geq 2$的单位$d$-区间图，这在图识别问题中并非直接成立——例如，对于$d$-轨道区间图的识别，从$d=2$到$d>2$的差距花了近20年才填补。我们的结果具有若干推论，包括识别$(x, \dots, x)$ $d$-区间图和深度$r$单位2-区间图对于每个$x\geq 11$和每个$r\geq 4$是NP完全的。