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

Abstract

To explain human performance on the Traveling Salesperson problem (TSP), MacGregor, Ormerod, and Chronicle (2000) proposed that humans construct solutions according to the steps described by their convex-hull algorithm. Focusing on tour length as the dependent variable, and using only random or semirandom point sets, the authors claimed empirical support for their model. In this paper we argue that the empirical tests performed by MacGregor et al. do not constitute support for the model, because they instantiate what Meehl (1997) coined "weak tests" (i.e., tests with a high probability of yielding confi rmation even if the model is false). To perform "strong" tests of the model, we implemented the algorithm in a computer program and compared its performance to that of humans on six point sets. The comparison reveals substantial and systematic differences in the shapes of the tours produced by the algorithm and human participants, for fi ve of the six point sets. The methodological lesson for testing TSP models is twofold: (1) Include qualitative measures (such as tour shape) as a dependent variable, and (2) use point sets for which the model makes "risky" predictions.

The Role of the Goal in Solving Hard Computational Problems: Do People Really Optimize?

“…In the case of E-TSP, it has been proposed that Gestalt principles of continuation and good form (Frisby & Stone, 2010) may be, at least partially, responsible for the solutions that participants find (Tak, Plaisier, & van Rooij, 2008). A hierarchical pyramid model that uses clustering produces solutions that closely match those found by participants (Graham et al, 2000).…”

Human Performance on the Traveling Salesman and Related Problems: A Review

“…In addition, performance on a contrived TSP with a single interior point was considerably better accounted for by a convex-hull model than a heuristic that combined crossing avoidance with a nearest neighbor decision rule (MacGregor, Chronicle, & Ormerod, 2004). In addition, specially constructed stimuli that represent strong tests of the crossing-avoidance hypothesis have been found to increase the rate of crossings from the typical rate of 6% of tours reported by van Rooij, Stege, and Schactman (2003), to more than 40% of tours (MacGregor, Chronicle, & Ormerod, 2004;Tak, Plaisier, & van Rooij, 2008).…”

“…Consequently, distinguishing between models may require tactics other than simple data fitting. One approach, suggested by Tak, Plaisier, and van Rooij (2008), is to devise and implement "strong" tests of models. While this is laudable for mature theories, a potential concern is that, at the present stage of development, the approach may eliminate all models.…”

Human Performance on Hard Non-Euclidean Graph Problems: Vertex Cover

“…These are problems that likely cannot be solved by polynomial-time algorithms 2 (Garey & Johnson, 1979). Studies of human performance on one such problem, the Euclidean Traveling Salesperson Problem (E-TSP), have indicated that humans may be able to find close to optimal solutions in near linear-time (Chronicle, MacGregor, Ormerod, & Burr, 2006;Dry, Lee, Vickers, & Hughes, 2006;Graham, Joshi, & Pizlo, 2000;MacGregor, Chronicle, & Ormerod, 2004;Kong & Schunn, 2007;Tak, Plaisier, & van Rooij, 2008). This seems to be in keeping with the best known approximation algorithms 3 that can also achieve near optimal results in close to linear time, as well as fast heuristics (Arora, 1997).…”