Convex hull or crossing avoidance? Solution heuristics in the traveling salesperson problem

Abstract

Untrained adults appear to have access to cognitive processes that allow them to perform well in the Euclidean version of the traveling salesperson problem (E-TSP). They do so despite the famous computational intractability of the problem, which stems from its combinatorial complexity. A current hypothesis is that humans' good performance is based on following a strategy of connecting boundary points in order (the convex hull hypothesis). Recently, an alternative has been proposed, that performance is governed by a strategy of avoiding crossings. We examined the crossing avoidance hypothesis from the perspectives of its capacity to explain existing data, its theoretical adequacy, and its ability to explain the results of three new experiments. In Experiment 1, effects on the solution quality of number of points versus number of interior points were compared. In Experiment 2, the distributions of observed paths were compared with those predicted from the two hypotheses. In Experiment 3, figural effects were varied to induce crossings. The results of the experiments were more consistent with the convex hull than with the crossing avoidance hypothesis. Despite its simplicity and intuitive appeal, crossing avoidance does not provide a complete alternative to the convex hull hypothesis. Further elucidation of human strategiesand heuristics for optimization problems such as the E-TSP will aid our understanding of how cognitive processes have adapted to the demands of combinatorial difficulty.

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

“…MacGregor and colleagues have proposed that a convex hull-based strategy can account for this fi nding (MacGregor & Ormerod, 1996;MacGregor et al, 2000MacGregor et al, , 2004. This raises two questions: Can the CH algorithm of MacGregor et al (2000) produce tours with crossings?…”

“…Second, aiming for a goal does not always guarantee you reach it. As also noted by MacGregor et al (2004) the "look ahead" required for crossing avoidance can be computationally too taxing at times, so that even a strategy that has as its primary goal to avoid crossings may yet produces tours with crossings. Be that as it may, we did not explicitly set out to test the crossing-avoidance hypothesis, but we thought it interesting to note that even a point set specifi cally constructed to "trick" people into making a crossing (MacGregor & Ormerod, 1996) often fails to do so.…”

Route planning with transportation network maps: an eye-tracking study

“…First, there are often dozens of stations and, given little advanced knowledge about the transportation network, the first phase of route planning, i.e., the establishment of the relationship between origin and destination, requires localising these two locations first. This is a visual search task which is not required in either TSP problems where all locations are targets (MacGregor & Ormerod, 1996;MacGregor et al, 2004) or in route choice from maps studies where origin and destination are either highlighted or learned before testing (Bailenson et al, 1998;Brunyé et al, 2010). Second, identifying and comparing route options and choosing the most viable path (the second and third phases in route planning: Brunyé et al, 2010) often requires taking into account transits between different lines or even transportation modes and these transits often have substantial impact on travel time or convenience (Ettema, Friman, Gärling, Olsson & Fujii, 2012;Raveau, Guo, Munoz, & Wilson, 2014).…”

“…The Convex hull method, for example, states that the first planning step produces a tour that encompasses all external dots. The remaining internal dots are added in later planning steps (MacGregor & Ormerod, 1996;MacGregor, Ormerod & Chronicle, 1999;MacGregor, Chronicle & Ormerod, 2004). The Hierarchical nearest neighbour method is a different planning heuristic which assumes that clusters of targets are established in the first step and routes are planned within these clusters.…”

“…In a second step the clusters are then linked together into a tour (Vickers, Lee, Dry & Hughes, 2003). Although the TSP task is different from situations in which users have to plan a route between only two locations (e.g., MacGregor et al, 2004;Wiener & Tenbrink, 2008), these studies demonstrate the use of strategies or heuristics to reduce the cognitive effort associated with route planning, i.e., the identification of and selection between alternative routes.…”