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Extended formulations for polygons. (English) Zbl 1290.68122
Summary: 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 $$O(\log n)$$, a result originating from work by A. Ben-Tal and A. Nemirovski [Math. Oper. Res. 26, No. 2, 193–205 (2001; Zbl 1082.90133)]. Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by M. Goemans [“Smallest compact formulation for the permutahedron” (2009), http://math.mit.edu/~goemans/PAPERS/permutahedron.pdf], and V. Kaibel and K. Pashkovich [Lect. Notes Comput. Sci. 6655, 287–300 (2011; Zbl 1341.90148)]. Second, we prove a lower bound of $$\sqrt{2n}$$ on the extension complexity of generic $$n$$-gons. Finally, we prove that there exist $$n$$-gons whose vertices lie on an $$O(n)\times O(n ^2)$$ integer grid with extension complexity $$\varOmega (\sqrt{n}/\sqrt{\log n})$$.

##### MSC:
 68U05 Computer graphics; computational geometry (digital and algorithmic aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
##### Keywords:
extended formulations; polygon; polytope; lower bound
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##### References:
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