Betweeness problem is well known NP-complete permutation problem. Given a family $M$ of triples $(a_i, a_j, a_k)$, the problem is to decide whether a permutation $\Phi$ of elements $a_1, a_2, ..., a_n$ exists which satisfies all betweeness constraints. For each triple $(a,b,c)$ in $M$, it holds that either $\Phi(a) \lt \Phi(b) \lt \Phi(c)$ or $\Phi(c) \lt \Phi(b) \lt \Phi(a)$.
Motivated partialy by this CS Theory post, I am interested in the relation between density of constraints and hardness of betweeness problem. I am looking for for a classification of problem's hardness based on the number of triples $|M|$ for these cases:
1-$|M|=O(\log n)$, 2-$|M|=o(n)$ , 3-$|M|=\Theta(n)$, 4-$|M|=\Omega(n^2)$, 5-$|M|=\Omega(n^3)$.
Please classify it to PTIME solvable, Quasi-polynomial solvable, Subexponential time solvable, and NP-complete.