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Erschienen in:
2017 | OriginalPaper | Buchkapitel
The Minimum Shared Edges Problem on Grid-Like Graphs
verfasst von : Till Fluschnik, Meike Hatzel, Steffen Härtlein, Hendrik Molter, Henning Seidler
Erschienen in: Graph-Theoretic Concepts in Computer Science
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Abstract
We study the \({\mathsf {NP}}\)-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route p paths from a start vertex to a target vertex in a given graph while using at most k edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number p of paths. On the contrary, we show that MSE remains \({\mathsf {NP}}\)-hard on subgraphs of bounded grids.
Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number p of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter k, p, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs.