Two fixed-parameter algorithms for Vertex Covering by Paths on Trees

Authors:
Jiong Guo;Rolf Niedermeier;Johannes Uhlmann
Affiliations:
Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Venue:
Information Processing Letters
Year:
2008

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Abstract

Vertex Covering by Paths on Trees with applications in machine translation is the task to cover all vertices of a tree T=(V,E) by choosing a minimum-weight subset of given paths in the tree. The problem is NP-hard and has recently been solved by an exact algorithm running in O(4^C@?|V|^2) time, where C denotes the maximum number of paths covering a tree vertex. We improve this running time to O(2^C@?C@?|V|). On the route to this, we introduce the problem Tree-like Weighted Hitting Set which might be of independent interest. In addition, for the unweighted case of Vertex Covering by Paths on Trees, we present an exact algorithm using a search tree of size O(2^k@?k!), where k denotes the number of chosen covering paths. Finally, we briefly discuss the existence of a size-O(k^2) problem kernel.