arXiv subjects
Publications and source records attributed to Kanstantsin Pashkovich.
It is well known that the permutahedron Pi_n has 2^n-2 facets. The Birkhoff polytope provides a symmetric extended formulation of Pi_n of size Theta(n^2). Recently, Goemans described a non-symmetric extended formulation of Pi_n of size Theta(n log(n)). In this paper, we prove that Omega(n^2) is a lower bound for the size of symmetric extended formulations of Pi_n.
In 1991, Yannakakis (J. Comput. System Sci., 1991) proved that no symmetric extended formulation for the matching polytope of the complete graph K_n 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 K_n. It was also conjectured in the paper mentioned above 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 K_n with log(n) (rounded down) 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 log(n) (rounded down). Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abtract that has appeared in the Proceedings of IPCO XIV at Lausanne, this paper does not only contain proofs that had been ommitted there, but it also presents slightly generalized and sharpened lower bounds.
An extended formulation of a polytope P is a polytope Q which can be projected onto P. Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes. The main known lower bounds on the minimum sizes of extended formulations for fixed polytope P (Yannakakis 1991) are closely related to the concept of nondeterministic communication complexity. We study the relative power and limitations of the bounds on several examples.
There are many examples of optimization problems whose associated polyhedra can be described much nicer, and with way less inequalities, by projections of higher dimensional polyhedra than this would be possible in the original space. However, currently not many general tools to construct such extended formulations are available. In this paper, we develop a framework of polyhedral relations that generalizes inductive constructions of extended formulations via projections, and we particularly elaborate on the special case of reflection relations. The latter ones provide polynomial size extended formulations for several polytopes that can be constructed as convex hulls of the unions of (exponentially) many copies of an input polytope obtained via sequences of reflections at hyperplanes. We demonstrate the use of the framework by deriving small extended formulations for the G-permutahedra of all finite reflection groups G (generalizing both Goeman's extended formulation of the permutahedron of size O(n log n) and Ben-Tal and Nemirovski's extended formulation with O(k) inequalities for the regular 2^k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).