Abstract
In this note, we consider the permutahedron, the convex hull of all permutations of \(\{1,2\ldots ,n\}\). We show how to obtain an extended formulation for this polytope from any sorting network. By using the optimal Ajtai–Komlós–Szemerédi sorting network, this extended formulation has \(\varTheta (n\log n)\) variables and inequalities. Furthermore, from basic polyhedral arguments, we show that this is best possible (up to a multiplicative constant) since any extended formulation has at least \(\varOmega (n \log n)\) inequalities. The results easily extend to the generalized permutahedron.
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References
Ajtai, M., Komlós, J., Szemerédi, E.: Sorting in \(C \log N\) parallel steps. Combinatorica 3, 1–19 (1983)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms (2001)
Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR 8, 1–48 (2010)
Fiorini, S., Kaibel, V., Pashkovich, K., Theis, D.O.: Combinatorial bounds on nonnegative rank and extended formulations. arxiv:1111.0444 (2012)
Fiorini, S., Massar, S., Pokutta, S., Tiwary, H.R., de Wolf, R.: Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds. In: Proceedings of the 44th ACM Symposium on Theory of Computing (STOC 2012) (2012)
Kaibel, V.: Extended formulations in combinatorial optimization. Optima 85, 2–7 (2011)
Pashkovich, K.: Tight lower bounds on the sizes of symmetric extensions of permutahedra and similar results. arxiv:0912.3446 (2009)
Paterson, M.S.: Improved sorting networks with \(O(\log N)\) depth. Algorithmica 5, 75–92 (1990)
Rado, R.: An inequality. J. Lond. Math. Soc. 27, 1–6 (1952)
Rothvoß, T.: Some \(0/1\) polytopes need exponential size extended formulations. Math. Program. (2012)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin (2003)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci. 43, 441–466 (1991)
Acknowledgments
In addition to Alberto Caprara, I would also like to thank Michele Conforti for a stimulating discussion at Oberwolfach, and Thomas Rothvoß for correcting calculations related to Theorem 3.
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Dedicated to Alberto Caprara who tragically died in a mountaineering accident in April 2012. The question answered here was asked by Alberto at the Oberwolfach meeting in Combinatorial Optimization in November 2008.
Supported by NSF contracts CCF-0829878 and 1115849, and by ONR Grant No. 0014-05-1-0148.
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Goemans, M.X. Smallest compact formulation for the permutahedron. Math. Program. 153, 5–11 (2015). https://doi.org/10.1007/s10107-014-0757-1
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DOI: https://doi.org/10.1007/s10107-014-0757-1