Jakub Kovac. Complexity of the path avoiding forbidden pairs problem revisited. Technical Report arXiv:1111.3996, arXiv.org, 2011.

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Let G = (V, E) be a directed acyclic graph with two distinguished vertices 
s, t and let F be a set of forbidden pairs of vertices. We say that a path 
in G is safe, if it contains at most one vertex from each pair {u, v} in F. 
Given G and F, the path avoiding forbidden pairs (PAFP) problem is to find a 
safe s-t path in G. We systematically study the complexity of different 
special cases of the PAFP problem defined according to the mutual positions 
of fobidden pairs. Fix one topological ordering of vertices; we say that 
pairs {u, v} and {x, y} are disjoint, if u, v < x, y, nested, if u < x, y < 
v, and halving, if u < x < v < y. The PAFP problem is known to be NP-hard in 
general or if no two pairs are disjoint; we prove that it remains NP-hard 
even when no two forbidden pairs are nested. On the other hand, if no two 
pairs are halving, the problem is known to be solvable in cubic time. We 
simplify and improve this result by showing an O(M(n)) time algorithm, where 
M(n) is the time to multiply two n \times n boolean matrices.