3,974 research outputs found
Extended Formulation Lower Bounds via Hypergraph Coloring?
Exploring the power of linear programming for combinatorial optimization
problems has been recently receiving renewed attention after a series of
breakthrough impossibility results. From an algorithmic perspective, the
related questions concern whether there are compact formulations even for
problems that are known to admit polynomial-time algorithms.
We propose a framework for proving lower bounds on the size of extended
formulations. We do so by introducing a specific type of extended relaxations
that we call product relaxations and is motivated by the study of the
Sherali-Adams (SA) hierarchy. Then we show that for every approximate
relaxation of a polytope P, there is a product relaxation that has the same
size and is at least as strong. We provide a methodology for proving lower
bounds on the size of approximate product relaxations by lower bounding the
chromatic number of an underlying hypergraph, whose vertices correspond to
gap-inducing vectors.
We extend the definition of product relaxations and our methodology to mixed
integer sets. However in this case we are able to show that mixed product
relaxations are at least as powerful as a special family of extended
formulations. As an application of our method we show an exponential lower
bound on the size of approximate mixed product formulations for the metric
capacitated facility location problem, a problem which seems to be intractable
for linear programming as far as constant-gap compact formulations are
concerned
A Semidefinite Hierarchy for Containment of Spectrahedra
A spectrahedron is the positivity region of a linear matrix pencil and thus
the feasible set of a semidefinite program. We propose and study a hierarchy of
sufficient semidefinite conditions to certify the containment of a
spectrahedron in another one. This approach comes from applying a moment
relaxation to a suitable polynomial optimization formulation. The hierarchical
criterion is stronger than a solitary semidefinite criterion discussed earlier
by Helton, Klep, and McCullough as well as by the authors. Moreover, several
exactness results for the solitary criterion can be brought forward to the
hierarchical approach. The hierarchy also applies to the (equivalent) question
of checking whether a map between matrix (sub-)spaces is positive. In this
context, the solitary criterion checks whether the map is completely positive,
and thus our results provide a hierarchy between positivity and complete
positivity.Comment: 24 pages, 2 figures; minor corrections; to appear in SIAM J. Opti
Efficient Relaxations for Dense CRFs with Sparse Higher Order Potentials
Dense conditional random fields (CRFs) have become a popular framework for
modelling several problems in computer vision such as stereo correspondence and
multi-class semantic segmentation. By modelling long-range interactions, dense
CRFs provide a labelling that captures finer detail than their sparse
counterparts. Currently, the state-of-the-art algorithm performs mean-field
inference using a filter-based method but fails to provide a strong theoretical
guarantee on the quality of the solution. A question naturally arises as to
whether it is possible to obtain a maximum a posteriori (MAP) estimate of a
dense CRF using a principled method. Within this paper, we show that this is
indeed possible. We will show that, by using a filter-based method, continuous
relaxations of the MAP problem can be optimised efficiently using
state-of-the-art algorithms. Specifically, we will solve a quadratic
programming (QP) relaxation using the Frank-Wolfe algorithm and a linear
programming (LP) relaxation by developing a proximal minimisation framework. By
exploiting labelling consistency in the higher-order potentials and utilising
the filter-based method, we are able to formulate the above algorithms such
that each iteration has a complexity linear in the number of classes and random
variables. The presented algorithms can be applied to any labelling problem
using a dense CRF with sparse higher-order potentials. In this paper, we use
semantic segmentation as an example application as it demonstrates the ability
of the algorithm to scale to dense CRFs with large dimensions. We perform
experiments on the Pascal dataset to indicate that the presented algorithms are
able to attain lower energies than the mean-field inference method
Intermediate integer programming representations using value disjunctions
We introduce a general technique to create an extended formulation of a
mixed-integer program. We classify the integer variables into blocks, each of
which generates a finite set of vector values. The extended formulation is
constructed by creating a new binary variable for each generated value. Initial
experiments show that the extended formulation can have a more compact complete
description than the original formulation.
We prove that, using this reformulation technique, the facet description
decomposes into one ``linking polyhedron'' per block and the ``aggregated
polyhedron''. Each of these polyhedra can be analyzed separately. For the case
of identical coefficients in a block, we provide a complete description of the
linking polyhedron and a polynomial-time separation algorithm. Applied to the
knapsack with a fixed number of distinct coefficients, this theorem provides a
complete description in an extended space with a polynomial number of
variables.Comment: 26 pages, 5 figure
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