Finding Two Edge-Disjoint Paths with Length Constraints

We consider the problem of finding, for two pairs (s1, t1) and (s2, t2) of vertices in an undirected graphs, an (s1, t1)-path P1 and an (s2, t2)-path P2 such that P1 and P2 share no edges and the length of each Pi satisfies Li, where Li ∈ {≤ ki, = ki, ≥ ki, ≤ ∞}. We regard k1 and k2 as parameters and investigate the parameterized complexity of the above problem when at least one of P1 and P2 has a length constraint (note that Li = “ ≤ ∞” indicates that Pi has no length constraint). For the nine different cases of (L1, L2), we obtain FPT algorithms for seven of them. Our algorithms uses random partition backed by some structural results. On the other hand, we prove that the problem admits no polynomial kernel for all nine cases unless NP ⊆ coNP/poly.