### Abstract

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints. We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge. Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.

Original language | English |
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Pages (from-to) | 1726-1735 |

Journal | Discrete Applied Mathematics |

Volume | 159 |

Issue number | 16 |

DOIs | |

Publication status | Published - 2011 |

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## Cite this

Darmann, A., Pferschy, U., Schauer, J., & Woeginger, G. J. (2011). Paths, trees and matchings under disjunctive constraints.

*Discrete Applied Mathematics*,*159*(16), 1726-1735. https://doi.org/10.1016/j.dam.2010.12.016