5,739 research outputs found
Uncapacitated Flow-based Extended Formulations
An extended formulation of a polytope is a linear description of this
polytope using extra variables besides the variables in which the polytope is
defined. The interest of extended formulations is due to the fact that many
interesting polytopes have extended formulations with a lot fewer inequalities
than any linear description in the original space. This motivates the
development of methods for, on the one hand, constructing extended formulations
and, on the other hand, proving lower bounds on the sizes of extended
formulations.
Network flows are a central paradigm in discrete optimization, and are widely
used to design extended formulations. We prove exponential lower bounds on the
sizes of uncapacitated flow-based extended formulations of several polytopes,
such as the (bipartite and non-bipartite) perfect matching polytope and TSP
polytope. We also give new examples of flow-based extended formulations, e.g.,
for 0/1-polytopes defined from regular languages. Finally, we state a few open
problems
Incompatible double posets and double order polytopes
In 1986 Stanley associated to a poset the order polytope. The close interplay
between its combinatorial and geometric properties makes the order polytope an
object of tremendous interest. Double posets were introduced in 2011 by
Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A
double poset is a finite set equipped with two partial orders. To a double
poset Chappell, Friedl and Sanyal (2017) associated the double order polytope.
They determined the combinatorial structure for the class of compatible double
posets. In this paper we generalize their description to all double posets and
we classify the 2-level double order polytopes.Comment: 11 pages, 3 figure
Sparse sum-of-squares certificates on finite abelian groups
Let G be a finite abelian group. This paper is concerned with nonnegative
functions on G that are sparse with respect to the Fourier basis. We establish
combinatorial conditions on subsets S and T of Fourier basis elements under
which nonnegative functions with Fourier support S are sums of squares of
functions with Fourier support T. Our combinatorial condition involves
constructing a chordal cover of a graph related to G and S (the Cayley graph
Cay(,S)) with maximal cliques related to T. Our result relies on two
main ingredients: the decomposition of sparse positive semidefinite matrices
with a chordal sparsity pattern, as well as a simple but key observation
exploiting the structure of the Fourier basis elements of G.
We apply our general result to two examples. First, in the case where , by constructing a particular chordal cover of the half-cube
graph, we prove that any nonnegative quadratic form in n binary variables is a
sum of squares of functions of degree at most , establishing
a conjecture of Laurent. Second, we consider nonnegative functions of degree d
on (when d divides N). By constructing a particular chordal
cover of the d'th power of the N-cycle, we prove that any such function is a
sum of squares of functions with at most nonzero Fourier
coefficients. Dually this shows that a certain cyclic polytope in
with N vertices can be expressed as a projection of a section
of the cone of psd matrices of size . Putting gives a
family of polytopes with LP extension complexity
and SDP extension complexity
. To the best of our knowledge, this is the
first explicit family of polytopes in increasing dimensions where
.Comment: 34 page
Theta Bodies for Polynomial Ideals
Inspired by a question of Lov\'asz, we introduce a hierarchy of nested
semidefinite relaxations of the convex hull of real solutions to an arbitrary
polynomial ideal, called theta bodies of the ideal. For the stable set problem
in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta
body of the graph. We prove that theta bodies are, up to closure, a version of
Lasserre's relaxations for real solutions to ideals, and that they can be
computed explicitly using combinatorial moment matrices. Theta bodies provide a
new canonical set of semidefinite relaxations for the max cut problem. For
vanishing ideals of finite point sets, we give several equivalent
characterizations of when the first theta body equals the convex hull of the
points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure
On cardinality constrained cycle and path polytopes
Given a directed graph D = (N, A) and a sequence of positive integers 1 <=
c_1 < c_2 < ... < c_m <= |N|, we consider those path and cycle polytopes that
are defined as the convex hulls of simple paths and cycles of D of cardinality
c_p for some p, respectively. We present integer characterizations of these
polytopes by facet defining linear inequalities for which the separation
problem can be solved in polynomial time. These inequalities can simply be
transformed into inequalities that characterize the integer points of the
undirected counterparts of cardinality constrained path and cycle polytopes.
Beyond we investigate some further inequalities, in particular inequalities
that are specific to odd/even paths and cycles.Comment: 24 page
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