The complexity of finding two disjoint paths with min-max objective function
Discrete Applied Mathematics
Easy problems for tree-decomposable graphs
Journal of Algorithms
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
Graph minors. XIII: the disjoint paths problem
Journal of Combinatorial Theory Series B
Journal of the ACM (JACM)
Improved bounds for the unsplittable flow problem
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The Quickest Multicommodity Flow Problem
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
Computing Disjoint Path with Lenght Constraints
WG '96 Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science
Journal of Computer and System Sciences
On the complexity of vertex-disjoint length-restricted path problems
Computational Complexity
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Improved bounds for the unsplittable flow problem
Journal of Algorithms
On the parameterized complexity of multiple-interval graph problems
Theoretical Computer Science
On problems without polynomial kernels
Journal of Computer and System Sciences
Planar Capacitated Dominating Set Is W[1]-Hard
Parameterized and Exact Computation
Paths of Bounded Length and Their Cuts: Parameterized Complexity and Algorithms
Parameterized and Exact Computation
Capacitated domination and covering: a parameterized perspective
IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
ACM Transactions on Algorithms (TALG)
Algorithmic lower bounds for problems parameterized by clique-width
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
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We study the parameterized complexity of two families of problems: the bounded length disjoint paths problem and the bounded length cut problem. From Menger's theorem both problems are equivalent (and computationally easy) in the unbounded case for single source, single target paths. However, in the bounded case, they are combinatorially distinct and are both NP-hard, even to approximate. Our results indicate that a more refined landscape appears when we study these problems with respect to their parameterized complexity. For this, we consider several parameterizations (with respect to the maximum length l of paths, the number k of paths or the size of a cut, and the treewidth of the input graph) of all variants of both problems (edge/vertex-disjoint paths or cuts, directed/undirected). We provide FPT-algorithms (for all variants) when parameterized by both k and l and hardness results when the parameter is only one of k and l. Our results indicate that the bounded length disjoint-path variants are structurally harder than their bounded length cut counterparts. Also, it appears that the edge variants are harder than their vertex-disjoint counterparts when parameterized by the treewidth of the input graph.