The roles of the convex hull and number of intersections upon performance on visually presented traveling salesperson problems

The planar Euclidean version of the Traveling Salesperson Problem (TSP) requires finding a tour of minimal length through a two-dimensional set of points. Despite the computational intractability of these problems, most empirical studies have found that humans are able to find good solutions. For this reason, understanding human performance on TSPs has the potential to offer insights into basic perceptual and cognitive decision making processes, as well as potentially informing theorizing in individual differences and intelligence. Through the convex hull hypothesis, previous researchers (MacGregor & Ormerod 1996; MacGregor, Ormerod, & Chronicle 1999; MacGregor, Ormerod, & Chronicle 2000; Ormerod & Chronicle 1999) have suggested people use a global-to-local solution process. We review the empirical evidence for and against this idea, before suggesting an alternative local-to-global solution process, based on the avoidance of intersections in constructing a tour. To compare these two competing approaches, we present the results of an experiment that measures the different effects the number of points on the convex hull and the number of potential intersections have on human performance. It is found that both have independent effects on the degree to which people deviate from the optimal solution, with performance worsening as more points are added to the convex hull and as fewer potential intersections are present. A measure of response uncertainty, capturing the degree to which different people produce the same types of solution, is found to be unaffected by the number of points on the convex hull but to increase as fewer potential intersections are present. A possible interpretation of these results in terms of a generative transformational theory of human visual perception is discussed. ∗Correspondence should be addressed to: Douglas Vickers, Department of Psychology, University of Adelaide, SA 5005, AUSTRALIA. Telephone: +61 8 8303 5662, Facsimile: +61 8 8303 3770, E-mail: douglas.vickers@psychology.adelaide.edu.au