Mathematics – Optimization and Control
Scientific paper
2006-11-17
Discrete Optimization 8 (2011), pp. 595-610
Mathematics
Optimization and Control
22 pages, revised and extended version of a previous version that has appeared under the same title in Proc. IPCO 2007
Scientific paper
10.1016/j.disopt.2011.07.001
The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the order of the subsets of the partition is irrelevant, since this kind of symmetry unnecessarily blows up the search tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving such symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the search tree, removes redundant parts of the tree produced by the above mentioned symmetry. The method relies on certain polyhedra, called orbitopes, which have been introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It does, however, not explicitly add inequalities to the model. Instead, it uses certain fixing rules for variables. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem.
Kaibel Volker
Peinhardt Matthias
Pfetsch Marc E.
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