Existence of interval arrangement satisfying constraints
Construct a graph in which each vertex corresponds to an endpoint of an interval. Then add the following edges:


If $\ell_i,r_i$ are the endpoints of $I_i$, then add the edge $\ell_i \to r_i$.
For a constraint $I_i &lt; I_j$, add the edge $r_i \to \ell_j$.
For a constraint $I_i \cap I_j \neq \emptyset$, add the edges $\ell_i \to r_j$ and $\ell_j \to r_i$.


Now check that there are no directed cycles. (This answer is inspired by an answer on stackexchange.)