13 research outputs found
Tight Lower Bounds on the Sizes of Symmetric Extensions of Permutahedra and Similar Results
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.Comment: corrected an error in the linear description of the permutahedron in
introductio
Smallest Compact Formulation for the Permutahedron
In this note, we consider the permutahedron, the convex hull of all permutations of {1,2…,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 Θ(nlogn) 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 Ω(nlogn) inequalities. The results easily extend to the generalized permutahedron.National Science Foundation (U.S.) (Contract CCF-0829878)National Science Foundation (U.S.) (Contract CCF-1115849)United States. Office of Naval Research (Grant 0014-05-1-0148
Equivariant semidefinite lifts and sum-of-squares hierarchies
A central question in optimization is to maximize (or minimize) a linear
function over a given polytope P. To solve such a problem in practice one needs
a concise description of the polytope P. In this paper we are interested in
representations of P using the positive semidefinite cone: a positive
semidefinite lift (psd lift) of a polytope P is a representation of P as the
projection of an affine slice of the positive semidefinite cone
. Such a representation allows linear optimization problems
over P to be written as semidefinite programs of size d. Such representations
can be beneficial in practice when d is much smaller than the number of facets
of the polytope P. In this paper we are concerned with so-called equivariant
psd lifts (also known as symmetric psd lifts) which respect the symmetries of
the polytope P. We present a representation-theoretic framework to study
equivariant psd lifts of a certain class of symmetric polytopes known as
orbitopes. Our main result is a structure theorem where we show that any
equivariant psd lift of size d of an orbitope is of sum-of-squares type where
the functions in the sum-of-squares decomposition come from an invariant
subspace of dimension smaller than d^3. We use this framework to study two
well-known families of polytopes, namely the parity polytope and the cut
polytope, and we prove exponential lower bounds for equivariant psd lifts of
these polytopes.Comment: v2: 30 pages, Minor changes in presentation; v3: 29 pages, New
structure theorem for general orbitopes + changes in presentatio
Lower bounds on matrix factorization ranks via noncommutative polynomial optimization
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples
Discrete Geometry (hybrid meeting)
A number of important recent developments in various branches of
discrete geometry were presented at the workshop, which took place in
hybrid format due to a pandemic situation. The presentations
illustrated both the diversity of the area and its strong connections
to other fields of mathematics such as topology, combinatorics,
algebraic geometry or functional analysis. The open questions abound
and many of the results presented were obtained by young researchers,
confirming the great vitality of discrete geometry
Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants
Combinatorial Optimization
This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th
Combinatorial generation via permutation languages. II. Lattice congruences
This paper deals with lattice congruences of the weak order on the symmetric
group, and initiates the investigation of the cover graphs of the corresponding
lattice quotients. These graphs also arise as the skeleta of the so-called
quotientopes, a family of polytopes recently introduced by Pilaud and Santos
[Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra,
associahedra, hypercubes and several other polytopes. We prove that all of
these graphs have a Hamilton path, which can be computed by a simple greedy
algorithm. This is an application of our framework for exhaustively generating
various classes of combinatorial objects by encoding them as permutations. We
also characterize which of these graphs are vertex-transitive or regular via
their arc diagrams, give corresponding precise and asymptotic counting results,
and we determine their minimum and maximum degrees. Moreover, we investigate
the relation between lattice congruences of the weak order and pattern-avoiding
permutations