Paths of Bounded Length and Their Cuts: Parameterized Complexity and Algorithms

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.