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Symmetry matters for sizes of extended formulations

by Kaibel, V.; Pashkovich, K.; Theis, D. O..

Series: 2011-24, Preprints

90C10 Integer programming

In 1991, Yannakakis [17] proved that no symmetric extended formulation for the matching polytope of the complete graph Kn with n nodes has a number of variables and constraints that is bounded subexponentially in n. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of Kn. It was also conjectured in [17] that 'asymmetry does not help much,' but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in Kn with blog nc edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length blog nc. Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract [12], this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.

polytope, projection, matching, cycle