105 research outputs found
Computing Optimal Morse Matchings
Morse matchings capture the essential structural information of discrete
Morse functions. We show that computing optimal Morse matchings is NP-hard and
give an integer programming formulation for the problem. Then we present
polyhedral results for the corresponding polytope and report on computational
results
Computing the Face Lattice of a Polytope from its Vertex-Facet Incidences
We give an algorithm that constructs the Hasse diagram of the face lattice of
a convex polytope P from its vertex-facet incidences in time O(min{n,m}*a*f),
where n is the number of vertices, m is the number of facets, a is the number
of vertex-facet incidences, and f is the total number of faces of P. This
improves results of Fukuda and Rosta (1994), who described an algorithm for
enumerating all faces of a d-polytope in O(min{n,m}*d*f^2) steps. For simple or
simplicial d-polytopes our algorithm can be specialized to run in time
O(d*a*f). Furthermore, applications of the algorithm to other atomic lattices
are discussed, e.g., to face lattices of oriented matroids.Comment: 14 pages; to appear in: Comput. Geom.; the new version contains some
minor extensions and corrections as well as a more detailed treatment of
oriented matroid
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
This paper deals with the computational complexity of conditions which
guarantee that the NP-hard problem of finding the sparsest solution to an
underdetermined linear system can be solved by efficient algorithms. In the
literature, several such conditions have been introduced. The most well-known
ones are the mutual coherence, the restricted isometry property (RIP), and the
nullspace property (NSP). While evaluating the mutual coherence of a given
matrix is easy, it has been suspected for some time that evaluating RIP and NSP
is computationally intractable in general. We confirm these conjectures by
showing that for a given matrix A and positive integer k, computing the best
constants for which the RIP or NSP hold is, in general, NP-hard. These results
are based on the fact that determining the spark of a matrix is NP-hard, which
is also established in this paper. Furthermore, we also give several complexity
statements about problems related to the above concepts.Comment: 13 pages; accepted for publication in IEEE Trans. Inf. Theor
A Compact Formulation for the Mixed-Norm Minimization Problem
Parameter estimation from multiple measurement vectors (MMVs) is a
fundamental problem in many signal processing applications, e.g., spectral
analysis and direction-of- arrival estimation. Recently, this problem has been
address using prior information in form of a jointly sparse signal structure. A
prominent approach for exploiting joint sparsity considers mixed-norm
minimization in which, however, the problem size grows with the number of
measurements and the desired resolution, respectively. In this work we derive
an equivalent, compact reformulation of the mixed-norm
minimization problem which provides new insights on the relation between
different existing approaches for jointly sparse signal reconstruction. The
reformulation builds upon a compact parameterization, which models the
row-norms of the sparse signal representation as parameters of interest,
resulting in a significant reduction of the MMV problem size. Given the sparse
vector of row-norms, the jointly sparse signal can be computed from the MMVs in
closed form. For the special case of uniform linear sampling, we present an
extension of the compact formulation for gridless parameter estimation by means
of semidefinite programming. Furthermore, we derive in this case from our
compact problem formulation the exact equivalence between the
mixed-norm minimization and the atomic-norm minimization. Additionally, for the
case of irregular sampling or a large number of samples, we present a low
complexity, grid-based implementation based on the coordinate descent method
Line Planning and Connectivity
The line planning problem in public transport deals with the
construction of a system of lines that is both attractive for the
passengers and of low costs for the operator.
In general, the computed line system should be connected, i.e., for each two stations there have to be a path that is covered by the lines.
This subproblem is a generalization of the well-known Steiner tree problem;
we call it the Steiner connectivity Problem. We discuss complexity of this problem, generalize the so-called
Steiner partition inequalities and give a transformation to the
directed Steiner tree problem. We show that directed models provide
tight formulations for the Steiner connectivity problem, similar as
for the Steiner tree problem
An Infeasible-Point Subgradient Method Using Adaptive Approximate Projections
We propose a new subgradient method for the minimization of nonsmooth convex
functions over a convex set. To speed up computations we use adaptive
approximate projections only requiring to move within a certain distance of the
exact projections (which decreases in the course of the algorithm). In
particular, the iterates in our method can be infeasible throughout the whole
procedure. Nevertheless, we provide conditions which ensure convergence to an
optimal feasible point under suitable assumptions. One convergence result deals
with step size sequences that are fixed a priori. Two other results handle
dynamic Polyak-type step sizes depending on a lower or upper estimate of the
optimal objective function value, respectively. Additionally, we briefly sketch
two applications: Optimization with convex chance constraints, and finding the
minimum l1-norm solution to an underdetermined linear system, an important
problem in Compressed Sensing.Comment: 36 pages, 3 figure
Sub-Exponential Lower Bounds for Branch-and-Bound with General Disjunctions via Interpolation
This paper investigates linear programming based branch-and-bound using
general disjunctions, also known as stabbing planes, for solving integer
programs. We derive the first sub-exponential lower bound (in the encoding
length of the integer program) for the size of a general branch-and-bound
tree for a particular class of (compact) integer programs, namely
for every . This is achieved by
showing that general branch-and-bound admits quasi-feasible monotone real
interpolation, which allows us to utilize sub-exponential lower-bounds for
monotone real circuits separating the so-called clique-coloring pair. One
important ingredient of the proof is that for every general branch-and-bound
tree proving integer-freeness of a product of two polytopes and
, there exists a closely related branch-and-bound tree for showing
integer-freeness of or one showing integer-freeness of . Moreover, we
prove that monotone real circuits can perform binary search efficiently
Vertex-Facet Incidences of Unbounded Polyhedra
How much of the combinatorial structure of a pointed polyhedron is contained
in its vertex-facet incidences? Not too much, in general, as we demonstrate by
examples. However, one can tell from the incidence data whether the polyhedron
is bounded. In the case of a polyhedron that is simple and "simplicial," i.e.,
a d-dimensional polyhedron that has d facets through each vertex and d vertices
on each facet, we derive from the structure of the vertex-facet incidence
matrix that the polyhedron is necessarily bounded. In particular, this yields a
characterization of those polyhedra that have circulants as vertex-facet
incidence matrices.Comment: LaTeX2e, 14 pages with 4 figure
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