Traveling Salesman Problem: A Foveating Pyramid Model

Abstract

We tested human performance on the Euclidean Traveling Salesman Problem using problems with 6-50 cities. Results confirmed our earlier findings that: (a) the time of solving a problem is proportional to the number of cities, and (b) the solution error grows very slowly with the number of cities. We formulated a new version of a pyramid model. The new model has an adaptive spatial structure, and it simulates visual acuity and visual attention. Specifically, the model solves the E-TSP problem sequentially by moving attention from city to city, the same way human subjects do. The model includes a parameter representing the magnitude of local search. This parameter allows modeling individual differences among the subjects. The computational complexity of the current implementation of the model is O(n 2 ), but this can most likely be improved to O[nlog(n)]. Simulation experiments demonstrated psychological plausibility of the new model.

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.…”

Individual Differences in Optimization Problem Solving: Reconciling Conflicting Results

“…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).…”

Development of the PEBL Traveling Salesman Problem Computerized Testbed

“…In contrast, some computerized solutions (e.g., Vickers et al, 2001;2004;Dry, Lee, Vickers, & Hughes, 2006) have used computerized methods that permit non-sequential solutions using a click-drag-release method, while most others (Graham, Joshi, & Pizlo, 2000;Pizlo, Stefanov, Saalweachter, Li, Haxhimusa, & Kropatsch, 2006;Chronicle et al, 2008;Acuña & Parada, 2010) have required sequential solutions.…”