1,519 research outputs found
Virtual Network Embedding via Decomposable LP Formulations: Orientations of Small Extraction Width and Beyond
The Virtual Network Embedding Problem (VNEP) considers the efficient
allocation of resources distributed in a substrate network to a set of request
networks. Many existing works discuss either heuristics or exact algorithms,
resulting in a choice between quick runtimes and quality guarantees for the
solutions. Recently, the first fixed-parameter tractable (FPT) approximation
algorithm for the VNEP with arbitrary request and substrate topologies has been
published by Rost and Schmid. This algorithm is based on a LP formulation and
is FPT in the newly introduced graph parameter extraction width (EW). It
therefore combines positive traits of heuristics and exact approaches: The
runtime is polynomial for instances with bounded EW, and the algorithm returns
approximate solutions with high probability.
We propose two extensions of this algorithm to optimize its runtime. Firstly,
we develop a new LP formulation related to tree decompositions. We show that
this LP formulation is always smaller, and that the resulting algorithm is FPT
in the new parameter extraction label width (ELW). We improve on two important
results by Rost and Schmid: For centrally rooted half-wheel topologies, the EW
scales linearly with request size, whereas the ELW is constant. Further, adding
any number of paths parallel to an existing edge increases the EW by at most
the maximum degree of the request. By contrast, the ELW only increases by at
most one. Lastly, we show that finding the minimal ELW is NP-hard.
Secondly, we consider the approach of partitioning the request into subgraphs
which are processed independently, yielding even smaller LP formulations. While
this algorithm may lead to higher ELW within the subgraphs, we show that this
increase is always smaller than the size of the inter-subgraph boundary. In
particular, the algorithm has zero additional cost when subgraphs are separated
by a single node.Comment: This work is a minor revision of the author's M.Sc. thesis in
Computer Science at Technische Universit\"at Berli
Convergence of Min-Sum Message Passing for Quadratic Optimization
We establish the convergence of the min-sum message passing algorithm for
minimization of a broad class of quadratic objective functions: those that
admit a convex decomposition. Our results also apply to the equivalent problem
of the convergence of Gaussian belief propagation
Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization
We propose an adaptive smoothing algorithm based on Nesterov's smoothing
technique in \cite{Nesterov2005c} for solving "fully" nonsmooth composite
convex optimization problems. Our method combines both Nesterov's accelerated
proximal gradient scheme and a new homotopy strategy for smoothness parameter.
By an appropriate choice of smoothing functions, we develop a new algorithm
that has the -worst-case
iteration-complexity while preserves the same complexity-per-iteration as in
Nesterov's method and allows one to automatically update the smoothness
parameter at each iteration. Then, we customize our algorithm to solve four
special cases that cover various applications. We also specify our algorithm to
solve constrained convex optimization problems and show its convergence
guarantee on a primal sequence of iterates. We demonstrate our algorithm
through three numerical examples and compare it with other related algorithms.Comment: This paper has 23 pages, 3 figures and 1 tabl
Service Chain and Virtual Network Embeddings: Approximations using Randomized Rounding
The SDN and NFV paradigms enable novel network services which can be realized
and embedded in a flexible and rapid manner. For example, SDN can be used to
flexibly steer traffic from a source to a destination through a sequence of
virtualized middleboxes, in order to realize so-called service chains. The
service chain embedding problem consists of three tasks: admission control,
finding suitable locations to allocate the virtualized middleboxes and
computing corresponding routing paths. This paper considers the offline batch
embedding of multiple service chains. Concretely, we consider the objectives of
maximizing the profit by embedding an optimal subset of requests or minimizing
the costs when all requests need to be embedded. Interestingly, while the
service chain embedding problem has recently received much attention, so far,
only non- polynomial time algorithms (based on integer programming) as well as
heuristics (which do not provide any formal guarantees) are known. This paper
presents the first polynomial time service chain approximation algorithms both
for the case with admission and without admission control. Our algorithm is
based on a novel extension of the classic linear programming and randomized
rounding technique, which may be of independent interest. In particular, we
show that our approach can also be extended to more complex service graphs,
containing cycles or sub-chains, hence also providing new insights into the
classic virtual network embedding problem
A counterexample to the Hirsch conjecture
The Hirsch Conjecture (1957) stated that the graph of a -dimensional
polytope with facets cannot have (combinatorial) diameter greater than
. That is, that any two vertices of the polytope can be connected by a
path of at most edges.
This paper presents the first counterexample to the conjecture. Our polytope
has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope
with 48 facets which violates a certain generalization of the -step
conjecture of Klee and Walkup.Comment: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by
referees. This version has been accepted in the Annals of Mathematic
The Refinement Calculus of Reactive Systems
The Refinement Calculus of Reactive Systems (RCRS) is a compositional formal
framework for modeling and reasoning about reactive systems. RCRS provides a
language which allows to describe atomic components as symbolic transition
systems or QLTL formulas, and composite components formed using three primitive
composition operators: serial, parallel, and feedback. The semantics of the
language is given in terms of monotonic property transformers, an extension to
reactive systems of monotonic predicate transformers, which have been used to
give compositional semantics to sequential programs. RCRS allows to specify
both safety and liveness properties. It also allows to model input-output
systems which are both non-deterministic and non-input-receptive (i.e., which
may reject some inputs at some points in time), and can thus be seen as a
behavioral type system. RCRS provides a set of techniques for symbolic
computer-aided reasoning, including compositional static analysis and
verification. RCRS comes with a publicly available implementation which
includes a complete formalization of the RCRS theory in the Isabelle proof
assistant
A Primal-Dual Algorithmic Framework for Constrained Convex Minimization
We present a primal-dual algorithmic framework to obtain approximate
solutions to a prototypical constrained convex optimization problem, and
rigorously characterize how common structural assumptions affect the numerical
efficiency. Our main analysis technique provides a fresh perspective on
Nesterov's excessive gap technique in a structured fashion and unifies it with
smoothing and primal-dual methods. For instance, through the choices of a dual
smoothing strategy and a center point, our framework subsumes decomposition
algorithms, augmented Lagrangian as well as the alternating direction
method-of-multipliers methods as its special cases, and provides optimal
convergence rates on the primal objective residual as well as the primal
feasibility gap of the iterates for all.Comment: This paper consists of 54 pages with 7 tables and 12 figure
Functional Nonlinear Sparse Models
Signal processing is rich in inherently continuous and often nonlinear
applications, such as spectral estimation, optical imaging, and
super-resolution microscopy, in which sparsity plays a key role in obtaining
state-of-the-art results. Coping with the infinite dimensionality and
non-convexity of these problems typically involves discretization and convex
relaxations, e.g., using atomic norms. Nevertheless, grid mismatch and other
coherence issues often lead to discretized versions of sparse signals that are
not sparse. Even if they are, recovering sparse solutions using convex
relaxations requires assumptions that may be hard to meet in practice. What is
more, problems involving nonlinear measurements remain non-convex even after
relaxing the sparsity objective. We address these issues by directly tackling
the continuous, nonlinear problem cast as a sparse functional optimization
program. We prove that when these problems are non-atomic, they have no duality
gap and can therefore be solved efficiently using duality and~(stochastic)
convex optimization methods. We illustrate the wide range of applications of
this approach by formulating and solving problems from nonlinear spectral
estimation and robust classification.Comment: Accepted for publication on the IEEE Transactions on Signal
Processin
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