Evaluating the importance of the convex hull in solving the Euclidean version of the traveling salesperson problem: reply to Lee and Vickers (2000)

Lee and Vickers (2000) suggest that the results of MacGregor and Ormerod (1996), showing that the response uncertainty to traveling salesperson problems (TSPs) increases with increasing numbers of nonboundary points, may have resulted as an artifact of constraints imposed in the construction of stimuli. The fact that similar patterns of results have been obtained for our "constrained" stimuli, for a stimulus constructed under different constraints, for 13 randomly generated stimuli, and for random and patterned 48-point problems provides empirical evidence that the results are not artifactual. Lee and Vickers further suggest that, even if not artifactual, the results are in principle limited to arrays of fewer than 50 points and that, beyond this, the total number of points and number of nonboundary points are "diagnostically equivalent." This claim seems to us incorrect, since arrays of any size can be constructed that will permit experimental tests of whether problem difficulty is influenced by the number of nonboundary points, or the total number of points, or both. We present a reanalysis of our original data using hierarchical regression analysis which indicates that both factors may influence problem complexity.