The traveling salesman problem: A hierarchical model

Abstract

Our review of prior literature on spatial information processing in perception, attention, and memory indicates that these cognitive functions involve similar mechanisms based on a hierarchical architecture. The present study extends the application of hierarchical models to the area of problem solving. First, we report results of an experiment in which human subjects were tested on a Euclidean traveling salesman problem (TSP) with 6 to 30 cities. The subject's solutions were either optimal or near-optimal in length and were produced in a time that was, on average, a linear function of the number of cities. Next, the performance of the subjects is compared with that of five representative artificial intelligence and operations research algorithms, that produce approximate solutions for Euclidean problems. None of these algorithms was found to be an adequate psychological model. Finally, we present a new algorithm for solving the TSP,which is based on a hierarchical pyramid architecture. The performance of this new algorithm is quite similar to the performance of the subjects.Processing of spatial information is an important cognitive ability. It is involved in such cognitive functions as perception, memory, attention, navigation, and problem solving. The multitude and diversity of these functions have often led students of cognition to assume the operation of several different and independent mechanisms (modules) for spatial information processing, each mechanism subserving its corresponding cognitive function. However, this account, recently referred to by the picturesque name "bag of tricks" (Ramachandran, 1990), should not be accepted until more parsimonious explanations are rejected. In fact, the results of prior research seem to suggest that visual perception, memory, and attention do involve similar mechanisms. Specifically, all these mechanisms are based on a hierarchical architecture This study was reported at annual meetings

Individual Differences in Optimization Problem Solving: Reconciling Conflicting Results

“…A number of theoretical accounts have suggested that how people solve TSPs is, in part, based on perceptual processes (Graham et al, 2000;MacGregor et al, 2000). It seems plausible that, for very simple problems, perceptual processing predominates but, as problem diffi culty increases, more analytical processes come into play.…”

“…Interestingly, human solutions are often better than those of the simpler heuristic procedures (Graham, Joshi, & Pizlo, 2000;MacGregor & Ormerod, 1996). In addition, while the computational times per node for successful heuristic procedures typically increase as a function of the number of nodes, it seems that human solution times per node remain constant.…”

“…In addition, while the computational times per node for successful heuristic procedures typically increase as a function of the number of nodes, it seems that human solution times per node remain constant. 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

“…2. TSP modelers and researchers may do well not to focus exclusively on replicating the high quality of human performance and consider successful replications as support for their models (see also Graham et al, 2000;Pizlo et al, 2006). Instead it may be better to evaluate TSP models on the basis of risky predictions; after all, if a model passes a test based on a risky prediction, then this counts as genuine support for the model (see also Meehl, 1997;Roberts & Pashler, 2000).…”

“…This observation has motivated researchers to set out to identify the human strategy for solving TSPs and implement it in a computational model. In this paper, we take a closer look at one such model, the Convex-hull (CH) algorithm model proposed by MacGregor, Ormerod, and Chronicle (2000), and its purported fi t to human performance on the TSP (for altogether different modeling attempts for TSP see Graham, Joshi, & Pizlo, 2000;and Pizlo et al, 2006). The CH model of MacGregor et al (2000) is a formal algorithmic elaboration on the convex-hull hypothesis put forth by MacGregor and Ormerod (1996;see van Rooij, Stege, & Schactman, 2003, for an overall assessment of the empirical support for this hypothesis) that proposes that people construct solutions to the TSP by fi rst perceiving (and mentally sketching) the boundary of the point set (called the convex hull) and then inserting one by one the interior points in the tour.…”