1,256 research outputs found
A Lagrangian relaxation approach to the edge-weighted clique problem
The -clique polytope is the convex hull of the node and edge incidence vectors of all subcliques of size at most of a complete graph on nodes. Including the Boolean quadric polytope as a special case and being closely related to the quadratic knapsack polytope, it has received considerable attention in the literature. In particular, the max-cut problem is equivalent with optimizing a linear function over . The problem of optimizing linear functions over has so far been approached via heuristic combinatorial algorithms and cutting-plane methods. We study the structure of in further detail and present a new computational approach to the linear optimization problem based on Lucena's suggestion of integrating cutting planes into a Lagrangian relaxation of an integer programming problem. In particular, we show that the separation problem for tree inequalities becomes polynomial in our Lagrangian framework. Finally, computational results are presented. \u
An exact mathematical programming approach to multiple RNA sequence-structure alignment
One of the main tasks in computational biology is the computation of
alignments of genomic sequences to reveal their commonalities. In case of DNA
or protein sequences, sequence information alone is usually sufficient to
compute reliable alignments. RNA molecules, however, build spatial
conformations—the secondary structure—that are more conserved than the actual
sequence. Hence, computing reliable alignments of RNA molecules has to take
into account the secondary structure. We present a novel framework for the
computation of exact multiple sequence-structure alignments: We give a graph-
theoretic representation of the sequence-structure alignment problem and
phrase it as an integer linear program. We identify a class of constraints
that make the problem easier to solve and relax the original integer linear
program in a Lagrangian manner. Experiments on a recently published benchmark
show that our algorithms has a comparable performance than more costly dynamic
programming algorithms, and outperforms all other approaches in terms of
solution quality with an increasing number of input sequences
Using underapproximations for sparse nonnegative matrix factorization
Nonnegative Matrix Factorization (NMF) has gathered a lot of attention in the last decade and has been successfully applied in numerous applications. It consists in the factorization of a nonnegative matrix by the product of two low-rank nonnegative matrices:. MªVW. In this paper, we attempt to solve NMF problems in a recursive way. In order to do that, we introduce a new variant called Nonnegative Matrix Underapproximation (NMU) by adding the upper bound constraint VW£M. Besides enabling a recursive procedure for NMF, these inequalities make NMU particularly well suited to achieve a sparse representation, improving the part-based decomposition. Although NMU is NP-hard (which we prove using its equivalence with the maximum edge biclique problem in bipartite graphs), we present two approaches to solve it: a method based on convex reformulations and a method based on Lagrangian relaxation. Finally, we provide some encouraging numerical results for image processing applications.nonnegative matrix factorization, underapproximation, maximum edge biclique problem, sparsity, image processing
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
A new graph-based method for pairwise global network alignment
<p>Abstract</p> <p>Background</p> <p>In addition to component-based comparative approaches, <it>network alignments </it>provide the means to study conserved network topology such as common pathways and more complex network motifs. Yet, unlike in classical sequence alignment, the comparison of networks becomes computationally more challenging, as most meaningful assumptions instantly lead to <it>NP</it>-hard problems. Most previous algorithmic work on network alignments is heuristic in nature.</p> <p>Results</p> <p>We introduce the graph-based <it>maximum structural matching </it>formulation for pairwise global network alignment. We relate the formulation to previous work and prove <it>NP</it>-hardness of the problem.</p> <p>Based on the new formulation we build upon recent results in computational structural biology and present a novel Lagrangian relaxation approach that, in combination with a branch-and-bound method, computes provably optimal network alignments. The Lagrangian algorithm alone is a powerful heuristic method, which produces solutions that are often near-optimal and – unlike those computed by pure heuristics – come with a quality guarantee.</p> <p>Conclusion</p> <p>Computational experiments on the alignment of protein-protein interaction networks and on the classification of metabolic subnetworks demonstrate that the new method is reasonably fast and has advantages over pure heuristics. Our software tool is freely available as part of the L<smcaps>I</smcaps>SA library.</p
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