Kanstantsin Pashkovich

Symmetry matters for the sizes of extended formulations

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

Constructing extended formulations from reflection relations

\lfloor\log n\rfloor

Tight Lower Bounds on the Sizes of Symmetric Extensions of Permutahedra and Similar Results

edges there are non-symmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulation of polynomial size exists. We furthermore prove similar statements for the polytopes associated with cycles of length

Enumeration of 2-Level Polytopes

\lfloor\log n\rfloor

Stable sets and graphs with no even holes

. Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot.

Symmetry Matters for Sizes of Extended Formulations

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 Goemans [6] extended formulation of the permutahedron of size O(n log n) and Ben-Tal and Nemirovskis [2] extended formulation with O(k) inequalities for the regular 2k-gon) and for Huffman-polytopes (the convex hulls of the weight-vectors of Huffman codes).

Hidden vertices in extensions of polytopes

It is well known that the permutahedron Πn has 2n − 2 facets. The Birkhoff polytope provides a symmetric extended formulation of Πn of size Θ(n2). Recently, Goemans described a non-symmetric extended formulation of Πn of size Θ(n log n). In this paper, we prove that Ω(n2) is a lower bound for the size of symmetric extended formulations of Πn. Moreover, we prove that the cardinality indicating polytope has the same tight lower bounds for the sizes of symmetric and nonsymmetric extended formulations as the permutahedron.

The projected faces property and polyhedral relations

We propose the first algorithm for enumerating all combinatorial types of 2-level polytopes of a given dimension d, and provide complete experimental results for d ≤ 6. Our approach is based on the notion of a simplicial core, that allows us to reduce the problem to the enumeration of the closed sets of a discrete closure operator, along with some convex hull computations and isomorphism tests.

Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-Genus Graphs

We develop decomposition/composition tools for efficiently solving maximum weight stable sets problems as well as for describing them as polynomially sized linear programs (using “compact systems”). Some of these are well-known but need some extra work to yield polynomial “decomposition schemes”. We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycle of length greater than three and a cap is a hole together with an additional node that is adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.

Uncapacitated flow-based extended formulations

In 1991, Yannakakis [J. Comput. System Sci., 43 (1991), pp. 441--466] proved that no symmetric extended formulation for the matching polytope of the complete graph