Abstract-An integrated method for clustering of QRS complexes is presented which includes basis function representation and self-organizing neural networks (NN's). Each QRS complex is decomposed into Hermite basis functions and the resulting coefficients and width parameter are used to represent the complex. By means of this representation, unsupervised self-organizing NN's are employed to cluster the data into 25 groups. Using the MIT-BIH arrhythmia database, the resulting clusters are found to exhibit a very low degree of misclassification (1.5%). The integrated method outperforms, on the MIT-BIH database, both a published supervised learning method as well as a conventional template cross-correlation clustering method.

A feedback neural network approach to communication routing problems is developed, with emphasis on multiple shortest path problems, with several requests for transmissions between distinct start and end nodes. The basic ingredients are a set of Potts neurons for each request, with interactions designed to minimize path lengths and prevent overloading of network arcs. The topological nature of the problem is conveniently handled using a propagator matrix approach. Although the constraints are global, the algorithmic steps are based entirely on local information, facilitating distributed implementations. In the polynomially solvable single-request case, the approach reduces to a fuzzy version of the Bellman-Ford algorithm. The method is evaluated for synthetic problems of varying sizes and load levels, by comparing to exact solutions from a branch-and-bound method, or to approximate solutions from a simple heuristic. With very few exceptions, the Potts approach gives high-quality legal solutions. The computational demand scales merely as the product of the numbers of requests, nodes, and arcs.

A Potts feedback neural network approach for finding good solutions to resource allocation problems with a non-fixed topology is presented. As a target application the airline crew scheduling problem is chosen. The topological complication is handled by means of a propagator defined in terms of Potts neurons. The approach is tested on artificial random problems tuned to resemble realworld conditions. Very good results are obtained for a variety of problem sizes. The computer time demand for the approach only grows like (number of flights) 3 . A realistic problem typically is solved within minutes, partly due to a prior reduction of the problem size, based on an analysis of the local arrival/departure structure at the single airports. 1 martin@thep.lu.se 2 carsten@thep.lu.se 3 bs@thep.lu.se• Each crew must follow a closed looprotation -starting and ending at the hub (see fig. 1).

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A novel method is presented and explored within the framework of Potts neural networks for solving optimization problems with a non-trivial topology, with the airline crew scheduling problem as a target application. The key ingredient to handle the topological complications is a propagator de®ned in terms of Potts neurons. The approach is tested on arti®cial problems generated with two real-world problems as templates. The results are compared against the properties of the corresponding unrestricted problems. The latter are subject to a detailed analysis in a companion paper (M. Lagerholm, C. Peterson, B. Soderberg, submitted to European Journal of Operational Research). Very good results are obtained for a variety of problem sizes. The computer time demand for the approach only grows like (number of¯ights) 3 . A realistic problem typically is solved within minutes, partly due to a prior reduction of the problem size, based on an analysis of the local arrival/departure structure at the single airports. To facilitate the reading for audiences not familiar with Potts neurons and mean ®eld (MF) techniques, a brief review is given of recent advances in their application to resource allocation problems. Ó