Minimum weight

About: Minimum weight is a research topic. Over the lifetime, 2002 publications have been published within this topic receiving 28244 citations.

Integration of structural optimization in the general design process for aircraft

Abstract: Formal mathematical optimization methods have been developed during the past 10-15 yr for the structural design of aircraft. Together with a reliable analysis program like finite element methods (FEM), they provide powerful tools for the structural design. They are efficient in at least two ways: 1) by producing designs that meet all specified requirements at minimum weight in one step and 2) relieving the engineer from a timeconsuming search for modifications that give better results, they allow more creative design modifications. MBB has developed an optimization code called MBB-LAGRANGE which uses mathematical programming and gradients to fulfill different constraints simultaneously. A method for solving large linear equation systems by an iterative method is described to show the effort which went into the program, formulating the physical problem in a very efficient mathematical way. Some examples depicting the successful application of the MBBLAGRANGE code are presented. This article closes with an outlook on how the optimization problem could be enlarged to include also the shape and size of airplanes.

Genetic algorithms with permutation-based representation for computing the distance of linear codes

Abstract: Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the Magma Computational Algebra System concerning the stored distances of some linear codes.

Minimum-weight cycle covers and their approximability

Abstract: A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L⊆N. We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2 - Ɛ for certain sets L. For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where n is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n). To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.

Minimum Cycle Bases and Their Applications

Abstract: Minimum cycle bases of weighted undirected and directed graphs are bases of the cycle space of the (di)graphs with minimum weight. We survey the known polynomial-time algorithms for their construction, explain some of their properties and describe a few important applications.

A deterministic algorithm for the distance and weight distribution of binary nonlinear codes

Abstract: Given a binary nonlinear code, we provide a deterministic algorithm to compute its weight and distance distribution, and in particular, its minimum weight and its minimum distance, which takes advantage of fast Fourier techniques. This algorithm's performance is similar to that of best-known algorithms for the average case, while it is especially efficient for codes with low information rate. We provide complexity estimates for several cases of interest.