Extended Formulations for Polygons

Abstract

The extension complexity of a polytope P is the smallest integer k such that P is the projection of a polytope Q with k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193-205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of √2n on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n)×O(n 2) integer grid with extension complexity Ω(√n/√log n). © 2012 Springer Science+Business Media, LLC.