Abstract
This paper revisits the classical edge-disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in P in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size 3. We present three results which use edge-separator based parameters to chart new islands of tractability in the complexity landscape of EDP. Our first and main result utilizes the fairly recent structural parameter tree-cut width (a parameter with fundamental ties to graph immersions and graph cuts): we obtain a polynomial-time algorithm for EDP on every graph class of bounded tree-cut width. Our second result shows that EDP parameterized by tree-cut width is unlikely to be fixed-parameter tractable. Our final, third result is a polynomial kernel for EDP parameterized by the size of a minimum feedback edge set in the graph.
Highlights
Edge-Disjoint Paths (EDP) is a fundamental routing graph problem: we are given a graph G and a set P containing pairs of vertices, and are asked to decide whether there is a set of |P| pairwise edge-disjoint paths in G connecting each pair in P
In cases where fixed-parameter algorithms are unlikely to exist, one can instead aim for so-called XP algorithms, i.e., algorithms which run in polynomial time for every fixed value of k
Given the parallels between EDP parameterized by tree-cut width and Vertex-Disjoint Paths (VDP) parameterized by treewidth, one would rightfully expect that the fixed-parameter tractability result on the latter [24] would be mirrored in the former case
Summary
Edge-Disjoint Paths (EDP) is a fundamental routing graph problem: we are given a graph G and a set P containing pairs of vertices (terminals), and are asked to decide whether there is a set of |P| pairwise edge-disjoint paths in G connecting each pair in P. Given the parallels between EDP parameterized by tree-cut width (an edge-separator based parameter) and VDP parameterized by treewidth (a vertex-separator based parameter), one would rightfully expect that the fixed-parameter tractability result on the latter [24] would be mirrored in the former case We rule this out by showing that EDP parameterized by tree-cut width is W[1]-hard [5, 7] and unlikely to be fixedparameter tractable; we obtain this lower-bound result even in the more restrictive setting of Simple EDP in Lemma 5. Having ruled out fixed-parameter algorithms for EDP parameterized by treecut width and in view of previous lower-bound results, one may ask whether it is even possible to obtain such an algorithm for any reasonable parameterization We answer this question positively by using the size of a minimum feedback edge set as a parameter.
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