Human performance on visually presented Traveling Salesman problems

Abstract

Little research has been carried out on human performance in optimization problems, such as the Traveling Salesman problem (TSP). Studies by Polivanova (1974, Voprosy Psikhologii, 4, 41-51) and by MacGregor and Ormerod (1996, Perception & Psychophysics, 58, 527-539) suggest that: (1) the complexity of solutions to visually presented TSPs depends on the number of points on the convex hull; and (2) the perception of optimal structure is an innate tendency of the visual system, not subject to individual differences. Results are reported from two experiments. In the first, measures of the total length and completion speed of pathways, and a measure of path uncertainty were compared with optimal solutions produced by an elastic net algorithm and by several heuristic methods. Performance was also compared under instructions to draw the shortest or the most attractive pathway. In the second, various measures of performance were compared with scores on Raven's advanced progressive matrices (APM). The number of points on the convex hull did not determine the relative optimality of solutions, although both this factor and the total number of points influenced solution speed and path uncertainty. Subjects' solutions showed appreciable individual differences, which had a strong correlation with APM scores. The relation between perceptual organization and the process of solving visually presented TSPs is briefly discussed, as is the potential of optimization for providing a conceptual framework for the study of intelligence.

Individual Differences in Optimization Problem Solving: Reconciling Conflicting Results

“…1 (Fall 2008) Vickers, Bovet, et al (2003) procedure, for example, may have allowed learning by providing feedback after each trial, and that the results may have refl ected individual differences in learning ability. However, while this could explain the results reported in Vickers, Bovet, et al and Vickers, Lee, et al (2003), it cannot explain those found in studies where feedback was not provided (Vickers et al, 2001;Vickers, Lee, Hughes, Dry & McMahon, 2006). In contrast, Vickers, Lee and colleagues have argued that the failure to fi nd individual differences by MacGregor and Ormerod (1996) and Ormerod and Chronicle (1999) is likely to have arisen from ceiling effects, due to the simple and highly constrained nature of the stimuli used (Vickers et al, 2001;Vickers et al, 2006).…”

“…This is the question of whether or not performance differs reliably across individuals. While Chronicle, MacGregor and Ormerod have consistently failed to fi nd individual differences in TSP performance (Chronicle, MacGregor & Ormerod, 2006;MacGregor & Ormerod, 1996), Lee, Vickers and colleagues have reported signifi cant differences, not only within TSP performance (Vickers, Bovet, Lee, & Hughes, 2003;Vickers et al, 2001), but between TSP performance and other cognitive tasks (Burns, Lee, & Vickers, 2006;Vickers, Mayo, Heitman, Lee, & Hughes, 2004). Table 1 provides a representative sample of the relevant fi ndings.…”

“…In other words, human solution times per problem increase in proportion to the number of nodes (Graham et al, 2000;Pizlo, Stefanov, Saalweachter, Li, Haxhimusa, & Kropatsch, 2006). Other generally agreed-upon fi ndings are that human solutions are typically close to optimal (Graham et al, 2000;MacGregor & Ormerod, 1996;van Rooij, Schactman, Kadlec, & Stege, 2006;Vickers, Butavicius, Lee, & Medvedev, 2001) and rarely self-intersect (MacGregor, Ormerod, & Chronicle, 2000;van Rooij, Stege, & Schactman, 2003;Vickers, Lee, Dry, & Hughes, 2003).…”

Some Tours are More Equal than Others: The Convex-Hull Model Revisited with Lessons for Testing Models of the Traveling Salesperson Problem

“…The reason is that the empirical tests performed by MacGregor et al were not strong tests of the CH model, because the tests were such that confi rmation was likely to be found, even if the model were false (Meehl, 1997). First of all, it is known that for many (randomly generated) point sets human and model performance is close to optimal (MacGregor et al, 2000;van Rooij, Schactman, Kadlec, & Stege, 2006;Vickers, Butavicius, Lee, & Medvedev, 2001), making it diffi cult to detect differences between human and model performance in terms of tour length. Put differently, any algorithm that produces close to optimal tours for random point sets will give a reasonably good fi t to human performance measured in terms of tour length.…”

Minimal Paths in the City Block: Human Performance on Euclidean and Non-Euclidean Traveling Salesperson Problems

“…One area in which this proposition has been examined closely is visually defined combinatorial optimization problems (e.g., Vickers et al, 2004), particularly traveling salesperson problems (TSPs; e.g., Polivanova, 1974;MacGregor & Ormerod, 1996;Vickers et al, 2001;Dry et al, 2006;Pizlo et al, 2006, Chronicle et al, 2008. In the standard closed tour Euclidean TSP, participants are shown an array of points, and asked to construct a path that passes through all points exactly once and returns to the starting point.…”