On the performance of self-organizing maps for the non-Euclidean Traveling Salesman Problem in the polygonal domain Jan Faigl ⇑ Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic article info Article history: Received 9 July 2010 Received in revised form 4 April 2011 Accepted 21 May 2011 Available online 27 May 2011 Keywords: Self-organizing map (SOM) Traveling Salesman Problem (TSP) Multi-goal path planning abstract In this paper, two state-of-the-art algorithms for the Traveling Salesman Problem (TSP) are examined in the multi-goal path planning problem motivated by inspection planning in the polygonal domain W. Both algorithms are based on the self-organizing map (SOM) for which an application in W is not typical. The first is Somhom’s algorithm, and the second is the Co-adaptive net. These algorithms are augmented by a simple approximation of the shortest path among obstacles in W. Moreover, the competitive and cooperative rules are modified by recent adaptation rules for the Euclidean TSP, and by proposed enhancements to improve the algorithms’ performance in the non-Euclidean TSP. Based on the modifica- tions, two new variants of the algorithms are proposed that reduce the required computa- tional time of their predecessors by an order of magnitude, therefore making SOM more competitive with combinatorial heuristics. The results show how SOM approaches can be used in the polygonal domain so they can provide additional features over the classical combinatorial approaches based on the complete visibility graph. Ó 2011 Elsevier Inc. All rights reserved. 1. Introduction The self-organizing map (SOM) also known as Kohonen’s unsupervised neural network, was first applied to the Traveling Salesman Problem (TSP) by Angéniol et al. [1] and Fort [19] in 1988. The TSP is probably the most famous combinatorial problem studied by the operational research community for more than five decades [10]. The problem is to find a route for visiting a given set of n cities (goals) so that the length of the route is minimized. In SOM, the output neurons are orga- nized into a unidimensional structure (cycle), and a solution is represented by synaptic weights that are adapted to the cities during the self-adaptation process. After the adaptation, the neurons are associated to the cities, and because of the unidi- mensional structure, the final city tour can be retrieved by traversing the cycle. The SOM adaptation schema for the TSP consists of two phases. A city is presented to the network, and a winner neuron is selected in the competitive phase. For a planar TSP where cities represent points in R 2 , the neurons’ weights can be consid- ered as points in the plane that are called nodes in this paper. So, the winner neuron is the node with the smallest distance to the city. Then, the adaptation can be described as a movement of the winner node together with its neighboring nodes to- ward the city. The adaptation is called a cooperative phase, as neighboring nodes also move, although by a shorter distance. After the complete presentation of all cities (one adaptation step), the procedure is repeated until the termination condition is not met, e.g., when the winner nodes are sufficiently close to the cities. Several SOM approaches have been proposed [9,8,6,7,36,34,3,35,4,11] in the history of the SOM application to the TSP. In these approaches, the adaptation rules have been modified [37,39], heuristics have been considered [30], and combinations 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.05.019 ⇑ Tel.: +420 224357224. E-mail address: xfaigl@labe.felk.cvut.cz Information Sciences 181 (2011) 4214–4229 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins