Extended Formulations for Packing and Partitioning Orbitopes

TL;DR

This paper presents compact extended formulations for packing and partitioning orbitopes, which are convex hulls of lexicographically sorted 0/1-matrices, aiding symmetry reduction in integer programming.

Contribution

It introduces new compact extended formulations for these orbitopes and simplifies their linear description using shifted-column inequalities.

Findings

Extended formulations are compact and efficient.

Shifted-column inequalities suffice for linear description.

Facilitates symmetry reduction in integer programming.

Abstract

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, 2008). These polytopes are the convex hulls of all 0/1-matrices with lexicographically sorted columns and at most, resp. 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 that basically shifted-column inequalities suffice in order to describe those orbitopes linearly.