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Extended Formulations for Packing and Partitioning Orbitopes

Dipartimento di Ingegneria dell'Impresa, Università di Roma "Tor Vergata," 00133 Rome, Italy
Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, 39106 Magdeburg, Germany
faenza{at}disp.uniroma2.it
kaibel{at}ovgu.de, http://www.math.uni-magdeburg.de/kaibel/

We give compact extended formulations for the packing and partitioning orbitopes (with respect to the full symmetric group) described and analyzed in Kaibel and Pfetsch [Kaibel, V., M. E. Pfetsch. 2008. Packing and partitioning orbitopes. Math. Programming, Ser. A 114(1) 1–36]. These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, respectively, exactly one 1-entry per row. They are important objects for symmetry reduction in certain integer programs. Using the extended formulations, we also derive a rather simple proof of the fact established in the paper mentioned above, that basically shifted-column inequalities suffice to describe those orbitopes linearly.

Key Words: polytope; symmetry; projection; shifted-column inequalities
History: Received: June 2, 2008; revision received: November 8, 2008;