Jakub Kováč. Complexity of the path avoiding forbidden pairs problem revisited. Discrete Applied Mathematics, 161(10-11):1506-1512. 2013.

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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}∈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 by the mutual positions of forbidden 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×n boolean matrices.