138 research outputs found
Tight Sum-of-Squares lower bounds for binary polynomial optimization problems
We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre
hierarchy. For binary polynomial optimization problems of degree and an
odd number of variables , we prove that levels of the
SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This
matches the recent upper bound result by Sakaue, Takeda, Kim and Ito.
Additionally, we study a conjecture by Laurent, who considered the linear
representation of a set with no integral points. She showed that the
Sherali-Adams hierarchy requires levels to detect the empty integer hull,
and conjectured that the SoS/Lasserre rank for the same problem is . We
disprove this conjecture and derive lower and upper bounds for the rank
Faster SDP hierarchy solvers for local rounding algorithms
Convex relaxations based on different hierarchies of linear/semi-definite
programs have been used recently to devise approximation algorithms for various
optimization problems. The approximation guarantee of these algorithms improves
with the number of {\em rounds} in the hierarchy, though the complexity of
solving (or even writing down the solution for) the 'th level program grows
as where is the input size.
In this work, we observe that many of these algorithms are based on {\em
local} rounding procedures that only use a small part of the SDP solution (of
size instead of ). We give an algorithm to
find the requisite portion in time polynomial in its size. The challenge in
achieving this is that the required portion of the solution is not fixed a
priori but depends on other parts of the solution, sometimes in a complicated
iterative manner.
Our solver leads to time algorithms to obtain the same
guarantees in many cases as the earlier time algorithms based on
rounds of the Lasserre hierarchy. In particular, guarantees based on rounds can be realized in polynomial time.
We develop and describe our algorithm in a fairly general abstract framework.
The main technical tool in our work, which might be of independent interest in
convex optimization, is an efficient ellipsoid algorithm based separation
oracle for convex programs that can output a {\em certificate of infeasibility
with restricted support}. This is used in a recursive manner to find a sequence
of consistent points in nested convex bodies that "fools" local rounding
algorithms.Comment: 30 pages, 8 figure
A Semidefinite Approach to the Cover Problem
We apply theta body relaxations to the -cover problem and show
polynomial time solvability for certain classes of graphs. In particular, we
give an effective relaxation where all --hole facets are valid, and
study its relation to an open question of Conforti et al. For the triangle free
problem, we show for that the theta body relaxations do not converge by
steps; we also prove for all an integrality gap of 2 for the
second theta body
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
Equivariant semidefinite lifts of regular polygons
Given a polytope P in , we say that P has a positive
semidefinite lift (psd lift) of size d if one can express P as the linear
projection of an affine slice of the positive semidefinite cone
. If a polytope P has symmetry, we can consider equivariant psd
lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest
families of polytopes with interesting symmetries are regular polygons in the
plane, which have played an important role in the study of linear programming
lifts (or extended formulations). In this paper we study equivariant psd lifts
of regular polygons. We first show that the standard Lasserre/sum-of-squares
hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus
yields an equivariant psd lift of size linear in N. In contrast we show that
one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1,
which is exponentially smaller than the psd lift of the sum-of-squares
hierarchy. Our construction relies on finding a sparse sum-of-squares
certificate for the facet-defining inequalities of the regular 2^n-gon, i.e.,
one that only uses a small (logarithmic) number of monomials. Since any
equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the
first example of a polytope with an exponential gap between sizes of
equivariant LP lifts and equivariant psd lifts. Finally we prove that our
construction is essentially optimal by showing that any equivariant psd lift of
the regular N-gon must have size at least logarithmic in N.Comment: 29 page
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
Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location
Metric facility location is a well-studied problem for which linear
programming methods have been used with great success in deriving approximation
algorithms. The capacity-constrained generalizations, such as capacitated
facility location (CFL) and lower-bounded facility location (LBFL), have proved
notorious as far as LP-based approximation is concerned: while there are
local-search-based constant-factor approximations, there is no known linear
relaxation with constant integrality gap. According to Williamson and Shmoys
devising a relaxation-based approximation for \cfl\ is among the top 10 open
problems in approximation algorithms.
This paper advances significantly the state-of-the-art on the effectiveness
of linear programming for capacity-constrained facility location through a host
of impossibility results for both CFL and LBFL. We show that the relaxations
obtained from the natural LP at levels of the Sherali-Adams
hierarchy have an unbounded gap, partially answering an open question of
\cite{LiS13, AnBS13}. Here, denotes the number of facilities in the
instance. Building on the ideas for this result, we prove that the standard CFL
relaxation enriched with the generalized flow-cover valid inequalities
\cite{AardalPW95} has also an unbounded gap. This disproves a long-standing
conjecture of \cite{LeviSS12}. We finally introduce the family of proper
relaxations which generalizes to its logical extreme the classic star
relaxation and captures general configuration-style LPs. We characterize the
behavior of proper relaxations for CFL and LBFL through a sharp threshold
phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599
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