74 research outputs found
Efficient Maximum-Likelihood Decoding of Linear Block Codes on Binary Memoryless Channels
In this work, we consider efficient maximum-likelihood decoding of linear
block codes for small-to-moderate block lengths. The presented approach is a
branch-and-bound algorithm using the cutting-plane approach of Zhang and Siegel
(IEEE Trans. Inf. Theory, 2012) for obtaining lower bounds. We have compared
our proposed algorithm to the state-of-the-art commercial integer program
solver CPLEX, and for all considered codes our approach is faster for both low
and high signal-to-noise ratios. For instance, for the benchmark (155,64)
Tanner code our algorithm is more than 11 times as fast as CPLEX for an SNR of
1.0 dB on the additive white Gaussian noise channel. By a small modification,
our algorithm can be used to calculate the minimum distance, which we have
again verified to be much faster than using the CPLEX solver.Comment: Submitted to 2014 International Symposium on Information Theory. 5
Pages. Accepte
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
On a Cardinality Constrained Multicriteria Knapsack Problem
We consider a variant of a knapsack problem with a fixed cardinality constraint. There are three objective functions to be optimized: one real-valued and two integer-valued objectives. We show that this problem can be solved efficiently by a local search. The algorithm utilizes connectedness of a subset of feasible solutions and has optimal run-time
Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems
Convex approximation sets for multiobjective optimization problems are a
well-studied relaxation of the common notion of approximation sets. Instead of
approximating each image of a feasible solution by the image of some solution
in the approximation set up to a multiplicative factor in each component, a
convex approximation set only requires this multiplicative approximation to be
achieved by some convex combination of finitely many images of solutions in the
set. This makes convex approximation sets efficiently computable for a wide
range of multiobjective problems - even for many problems for which (classic)
approximations sets are hard to compute.
In this article, we propose a polynomial-time algorithm to compute convex
approximation sets that builds upon an exact or approximate algorithm for the
weighted sum scalarization and is, therefore, applicable to a large variety of
multiobjective optimization problems. The provided convex approximation quality
is arbitrarily close to the approximation quality of the underlying algorithm
for the weighted sum scalarization. In essence, our algorithm can be
interpreted as an approximate variant of the dual variant of Benson's Outer
Approximation Algorithm. Thus, in contrast to existing convex approximation
algorithms from the literature, information on solutions obtained during the
approximation process is utilized to significantly reduce both the practical
running time and the cardinality of the returned solution sets while still
guaranteeing the same worst-case approximation quality. We underpin these
advantages by the first comparison of all existing convex approximation
algorithms on several instances of the triobjective knapsack problem and the
triobjective symmetric metric traveling salesman problem
Art Spiegelman, MetaMaus, Pantheon, 2011 (version française : Flammarion, 2012)Art Spiegelman, Co-Mix, A Retrospective of Comics, Graphics, and Scraps / Une rétrospective de bandes dessinées, graphisme et débris divers, Flammarion, 2012
Il n’est plus besoin à notre époque de présenter Art Spiegelman et son oeuvre-maîtresse, Maus, seule bande dessinée à avoir jamais obtenu un prix Pulitzer et devenue en un quart de siècleun des plus grands récits autour de la Shoah. Deux ouvrages récemment parus permettent de découvrir la genèse de cet ouvrage unique en son genre et de l’œuvre au sens large de son auteur. Publié à l’automne 2011 chez Pantheon, MetaMaus est un objet singulier. Sous-titré « A Look Inside a Modern Classic », il ..
Using Scalarizations for the Approximation of Multiobjective Optimization Problems: Towards a General Theory
We study the approximation of general multiobjective optimization problems
with the help of scalarizations. Existing results state that multiobjective
minimization problems can be approximated well by norm-based scalarizations.
However, for multiobjective maximization problems, only impossibility results
are known so far. Countering this, we show that all multiobjective optimization
problems can, in principle, be approximated equally well by scalarizations. In
this context, we introduce a transformation theory for scalarizations that
establishes the following: Suppose there exists a scalarization that yields an
approximation of a certain quality for arbitrary instances of multiobjective
optimization problems with a given decomposition specifying which objective
functions are to be minimized / maximized. Then, for each other decomposition,
our transformation yields another scalarization that yields the same
approximation quality for arbitrary instances of problems with this other
decomposition. In this sense, the existing results about the approximation via
scalarizations for minimization problems carry over to any other objective
decomposition -- in particular, to maximization problems -- when suitably
adapting the employed scalarization.
We further provide necessary and sufficient conditions on a scalarization
such that its optimal solutions achieve a constant approximation quality. We
give an upper bound on the best achievable approximation quality that applies
to general scalarizations and is tight for the majority of norm-based
scalarizations applied in the context of multiobjective optimization. As a
consequence, none of these norm-based scalarizations can induce approximation
sets for optimization problems with maximization objectives, which unifies and
generalizes the existing impossibility results concerning the approximation of
maximization problems
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