The directed 2-linkage problem with length constraints

Abstract

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1, s2, t1, t2 whether D contains a pair of arc-disjoint paths P1, P2 such that Pi is an (si, ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [7]. Recently it was shown [9] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair constants k1, k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1, P2 such that Pi is an (si, ti)-path and the length of Pi is no more than d(si, ti) + ki, for i = 1, 2, where d(si, ti) denotes the length of the shortest (si, ti)path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1, P2 to the weak 2-linkage problem where each path Pi has length at most d(si, ti) + c log 1+ǫ n for some constant c. We also prove that the weak 2-linkage problem remains NPcomplete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.

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“…We prove that the problem of finding a compatible s-t path is fixed-parameter tractable when parameterized by the treecut-width of the graph. More precisely, the problem can be solved in time k O (k 2 ) • n 3 where k denotes the treecut-width.…”

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“…By the definition of edges of G, we get that P 1 and P 2 are two T -compatible edgedisjoint paths in G such that P i is from s i to t i for i = 1, 2. We can construct a graph G in O(|E| 3 ) time and find a path from (s…”

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“…We also show that this problem is W [1]-hard parameterized by k, which means that we cannot hope to remove the dependency in k in the exponent. However, we can extend the previous result to the case where paths are not required to be shortest paths, but of length at most d(s i , t i ) + C, where C is a fixed constant. )…”

The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

“…This problem is currently a very active topic and new algorithms have been found very recently for several variants in the case r = 2. Polynomial algorithms have been developed by Gottschau et al [20] and by Kobayashi and Sako [31] for undirected graphs with non-negative weighted edges and by Bang-Jensen et al [3] in the directed unweighted case where paths do not have to be shortest but have bounded lengths. The complexity of the problem is still open for r ≥ 3.…”