Traveling Salesperson Problem with Simple Obstacles: The Role of Multidimensional Scaling and the Role of Clustering

Abstract

The objective of the traveling salesperson problem (TSP) is to find a shortest tour through all nodes in a graph. Euclidean TSPs are traditionally represented with “cities” placed on a 2D plane. When straight line obstacles are added to the plane, a tour has to visit all cities while going around obstacles. The resulting problem with obstacles remains metric, but is not Euclidean because the shortest paths are no longer straight lines. We first revise a previous version of multiresolution graph pyramid by modifying the hierarchical clustering stage. Next, we describe two new experiments with human subjects. In the first experiment, the effect of the length of obstacles on the quality of tours produced by subjects was tested with three problem sizes. Long obstacles affect the tours to a greater degree than short obstacles, but long obstacles create obvious clusters and limit the ways in which the tours can be produced. In the second experiment we evaluated the degree to which Multidimensional Scaling (MDS) can compensate for the presence of obstacles. The results show that although MDS approximation can compensate to a large degree for the presence of obstacles, it cannot fully account for human performance. This fact suggests that mental representation of a TSP with obstacles is not Euclidean. Instead, it is likely to be based on hierarchical clustering in which pairwise distances represent the shortest paths around obstacles.