The roles of the convex hull and number of intersections ... - CiteSeerX

The roles of the convex hull and number of intersections upon performance on visually presented traveling salesperson problems. Douglas Vickers∗ Michael D. Lee

Matthew Dry Peter Hughes

Department of Psychology University of Adelaide

Abstract The planar Euclidean version of the Traveling Salesperson Problem (TSP) requires finding a tour of minimal length through a two-dimensional set of points. Despite the computational intractability of these problems, most empirical studies have found that humans are able to find good solutions. For this reason, understanding human performance on TSPs has the potential to offer insights into basic perceptual and cognitive decision making processes, as well as potentially informing theorizing in individual differences and intelligence. Through the convex hull hypothesis, previous researchers (MacGregor & Ormerod 1996; MacGregor, Ormerod, & Chronicle 1999; MacGregor, Ormerod, & Chronicle 2000; Ormerod & Chronicle 1999) have suggested people use a global-to-local solution process. We review the empirical evidence for and against this idea, before suggesting an alternative local-to-global solution process, based on the avoidance of intersections in constructing a tour. To compare these two competing approaches, we present the results of an experiment that measures the different effects the number of points on the convex hull and the number of potential intersections have on human performance. It is found that both have independent effects on the degree to which people deviate from the optimal solution, with performance worsening as more points are added to the convex hull and as fewer potential intersections are present. A measure of response uncertainty, capturing the degree to which different people produce the same types of solution, is found to be unaffected by the number of points on the convex hull but to increase as fewer potential intersections are present. A possible interpretation of these results in terms of a generative transformational theory of human visual perception is discussed. ∗ Correspondence should be addressed to: Douglas Vickers, Department of Psychology, University of Adelaide, SA 5005, AUSTRALIA. Telephone: +61 8 8303 5662, Facsimile: +61 8 8303 3770, E-mail: [email protected]

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Introduction Problem-solving is a central cognitive activity and an essential ingredient in human intelligence. For reasons of tractability, most studies of human problem-solving have employed tasks characterized by well-defined initial and terminating states that must be linked by a systematic, finite series of steps. In contrast, there is an important class of real-world problems for which no algorithm can be guaranteed to produce a definitive solution within a reasonable time. Recently, there has been a growing interest in human performance on a well-known example of such computationally infeasible problems, the so-called Traveling Salesperson Problem (TSP), traditionally referred to as the Traveling Salesman Problem (Graham, Joshi, & Pizlo 2000; Lee & Vickers 2000; MacGregor & Ormerod 1996; MacGregor, Ormerod, & Chronicle 1999; MacGregor, Ormerod, & Chronicle 2000; Vickers, Butavicius, Lee, & Medvedev 2001). The classic planar Euclidean TSP can be formulated as follows: Given n interconnected cities (represented by points in the plane), devise an itinerary that will visit each city exactly once, return to the initial city, and keep the total distance traveled to a minimum. To arrive at a definitive solution means considering (n − 1)!/2 itineraries. Although this is feasible when n is a small number, such as 5, and only 12 different routes have to be considered, the combinatorial explosion of possibilities means that it rapidly becomes impossible as n becomes even moderately large. The TSP is one of an important class of so-called NP-complete problems, for which it is believed there is no algorithm that can be guaranteed to arrive at a definitive solution within a reasonable, polynomial time (i.e., in a time proportional to nk , where k is some integer) (Lawler, Lenstra, Rinooy Kan, & Schmoys 1985; Wilf 1986). Despite the difficulty of such problems, most studies have found that human performance on visually presented TSP tasks compares well with the most successful computational procedures (e.g., MacGregor & Ormerod 1996; Vickers et al. 2001). For this reason, understanding how humans perform TSP tasks is an important problem for psychology. It has potential to offer insights into basic perceptual and cognitive decision making processes, as well as to inform theorizing in individual differences and intelligence research. In this paper, we compare two approaches to understanding how humans solve visually presented TSPs. The first assumes a global-to-local processing strategy, in which a rough global reference frame is first established, into which local information is then integrated. We focus on a well-developed specific instantiation of this approach, known as the convex hull hypothesis (MacGregor & Ormerod 1996), which assumes that solutions are guided by the boundary points in the stimulus array. We review the theoretical and empirical evidence for and against this hypothesis, before suggesting an alternative local-to-global approach. Under this alternative approach, solutions are assumed to be primarily guided by local processing constraints, such as the avoidance of intersections in constructing a tour. To compare these two approaches empirically, we present an experiment that measures the different effects the number of points on the convex hull and the number of potential intersections have on human performance.

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Global-to-Local Processing MacGregor and Ormerod (1996) found that the group performance of observers in two experiments surpassed that of a suite of computational heuristics and was so close to the best known solutions that there were no individual differences and there was zero correlation between performance across different problems. On this basis, they concluded that the consistent ability to arrive at near-optimal solutions argued for the operation of a common underlying process or set of processes and suggested that such processes might correspond to natural organizing tendencies of the visual system. This last conclusion is consistent with an experiment by Pomerantz (1981), who found that observers, asked to connect up the dots in arrays, to illustrate how they perceived the arrays, frequently connected the dots into the shortest possible path. The same conclusion is implied by the results of Ormerod and Chronicle (1999, Experiment 1), who found that observers’ ratings of goodness of figure were inversely related both to the number of interior points and to the amount by which solution pathways exceeded the optimum length. This conclusion is also supported by Vickers et al. (2001), who found that one group of 18 observers, instructed to draw the most aesthetically pleasing pathways, tended to produce the same solutions as a similar group that was explicitly instructed to produce the shortest pathways. If performance on visually presented TSPs is mediated by natural perceptual tendencies, then one possibility is that observers make use of an innate perceptual preference for organizing stimulus elements into convex figures, as suggested by Gestalt analyses of figure-ground differentiation in perception (Palmer 1999, pp. 282-283), and as volunteered by some observers in the early experiments of Polivanova (1974). Evidence that structural regularity in a TSP array facilitates performance has also been presented by MacGregor, Ormerod, and Chronicle (1999). An automatic preference for convexity would make it natural for observers in TSP tasks to use the convex hull bounding an array as a first step in arriving at a solution. This strategy would allow observers to exploit the principle, enunciated by Flood (1956), that the optimal solution will always connect adjacent points on the convex hull in sequence (even though it also passes through interior points in the process). Because of this principle, the convex hull forms the basis for a number of heuristic algorithms for solving TSPs (e.g., Golden, Bodin, Doyle, & Stewart 1980; Norback & Love 1977). As Ormerod and Chronicle (1999) point out, a procedure of this kind can be characterized as a global-to-local process, similar to that suggested by Sanocki (1991, 1993) as underlying object identification. According to this view, observers first establish a rough global reference frame, into which local information is then integrated. Such a procedure has the advantage that it allows the processing of the interior points without the additional computation involved in dealing with points on the convex hull. It follows that optimality of performance should be inversely related to the number of interior points. Evidence in Favor of the Convex Hull Hypothesis MacGregor and Ormerod (1996) carried out two experiments to test the idea that observers use the convex hull in this way. The first experiment employed six basic 10-point arrays plus a seventh so-called Dantzig problem, chosen because it represents a benchmark test that many relatively effective heuristics fail to solve (Dantzig, Fulkerson, & Johnson 1959). The seven arrays differed in the number of points on the convex hull (and hence

TSP Performance 4 in the number of interior, nonboundary points). The number of points on the convex hull ranged from 1 to 9. MacGregor and Ormerod (1996) found that there was a correlation of r = .94 between the number of interior points and the number of different solutions generated. There was also a correlation of r = .93 for the six basic problems between the number of interior points and a response uncertainty measure that summarized the frequency with which different participants connected the same nodes in their solutions. Of the 315 solutions, 98% adhered to the principle of connecting the adjacent boundary points in order. In addition, observers’ solutions had significantly fewer indentations than would be expected from availability. This was interpreted as consistent with the hypothesis that, not only do observers have a preference for connecting points on the convex hull in sequence, but they also have a preference for connecting adjacent points on the boundary to each other. In MacGregor and Ormerod’s first experiment, the maximum possible number of indentations and the number of interior points were highly correlated, and their second experiment was undertaken to distinguish the effects of these two variables. This experiment employed seven different 20-point arrays, with the number of interior points ranging from 4 up to 16, in steps of 2, and with the corresponding maximum numbers of indentations being 4, 6, 8, 10, 8, 6, and 4. Unlike experiment 1, the number of different solutions failed to correlate significantly with either the number of interior points or the maximum number of indentations. On the other hand, measures of response uncertainty for the seven problems again showed a (just) significant correlation with the number of interior points (r = .79). Similarly, only one out of 140 solutions failed to adhere to the principle of connecting the boundary points in order of adjacency, and the average number of indentations was about half of that expected by chance. Re-analyzing results for 13 problems, obtained by Lee (1985), MacGregor and Ormerod (1996) found that there was also a significant correlation (r = .84) between the number of interior points and the mean percentage by which observers’ total path lengths exceeded the independently calculated optimal. In addition, MacGregor, Ormerod, and Chronicle (2000) found that only 11 out of 455 solutions had intersections. According to MacGregor, Ormerod, and Chronicle, “[t]hese consistent elements argue that a relatively uniform process underlies performance. Furthermore, the results suggest the general form of that process to be similar to a convex hull approach” (2000, p. 1184). Evidence of a less direct kind in favor of a global-to-local model of the TSP solution procedure has also been advanced, on the basis of a somewhat different approach, by Ormerod and Chronicle (1999). In each of a time-limited, a time-unlimited, and a primed task (Experiments 2, 3, and 4), these authors found that the proportion of errors made and the time, taken by observers to decide whether or not a presented solution was optimal, both decreased as a function of the amount by which the solution exceeded the optimum. These findings are interpreted as consistent with the view that observers use global properties to identify a perceptually good structure rapidly and then complete this subtour using local cues. In agreement with this view, MacGregor et al. (1999) reported that observers’ path lengths for a 48-point array were unaffected by the shape or significance of two alternative frames, in which the array was presented, and that both the number of different solutions and the mean percentage above optimal were much greater for a random array than for a regular grid structure. Of more specific relevance to the convex hull hypothesis, MacGregor et al. (1999) also found that, when their number was held constant, solution performance (measured by the number of observers finding the

TSP Performance 5 optimal solution) was affected by whether the interior points were distributed in a regular or an irregular arrangement and by whether they were located towards the center or the periphery of the array as a whole. On the basis of the above findings, MacGregor et al. (2000) have proposed a computational model of the way in which observers solve TSPs. This model begins by sketching the connections (arcs) between adjacent vertices of the convex hull. A starting point and a direction (clockwise or anticlockwise) are selected randomly. If the starting point is not on the hull, the model uses either the cheapest insertion or the largest angle criterion to connect it with the end point of the closest boundary arc which is in the direction of travel. This becomes the current node. Otherwise, if the starting point is on the hull, the model applies the insertion criterion to check whether the current boundary arc has an interior point that is closer to it than to any other boundary arc. In this way, the model continues round the convex hull and the intervening indentation arcs that it creates, testing for interior points that are closest to each current arc in turn. Thus, the model imitates the way in which the human observer works sequentially from point to point. Moreover, like the human observer, but again unlike the standard convex hull heuristic program, the model is also capable of producing a variety of different solutions to a single problem. MacGregor et al. (2000) compared the performance of this model with that of human observers, as recorded in the previous studies of MacGregor and Ormerod (1996), Lee (1985), Krolak, Felts, and Marble (1971), and MacGregor et al. (1999), which investigated performance on irregular arrays containing from 10 up to 100 nodes. Using percentage difference between model performance and path lengths of human observers as a criterion of goodness of fit, MacGregor et al. (2000) found that the solutions produced by the model conformed closely to those produced by the human observers, with the cheapest insertion criterion faring rather better overall than the largest angle rule. However, the model tended to produce a narrower range of solutions than did human observers, and the worst solutions generated by the model tended to be better than the worst solutions produced by the human observers. In sum, this approach to understanding human performance on visually presented TSPs argues that the observer exploits spontaneous perceptual organizing principles to arrive at a subtour, which is then used to guide the completion of the problem in a serial and more local manner. In particular, it is argued, the observer makes use of observers’ preference for convex figures to establish the convex hull as a preliminary subtour. Both the general and the more specific interpretations of the global-to-local processing hypothesis have a number of merits and advantages. First, the convex hull hypothesis conforms to Flood’s (1956) useful heuristic principle that the optimal solution will connect points on the convex hull in sequence (passing through interior points to do so). Second, this hypothesis appears to make correct but non-obvious predictions concerning the influence of the number of interior points on solution quality, as well as a number of more obvious and logically interconnected predictions, such as the order in which points on the convex hull are connected, the lower than expected number of indentations, and the general avoidance of solutions with lines that cross. Third, the more general hypothesis is consistent with the finding that performance is influenced by the distribution and location of interior points, with the shorter time taken by observers to identify completed optimal solutions than to identify non-optimal pathways, with the superior performance on regular than on irregular arrays, and with the apparent impenetrability of the hypothesized process to cognitive factors, provided

TSP Performance 6 by embedding arrays in a meaningful background. Fourth, the more specific, convex hull hypothesis has been implemented in a relatively simple, yet flexible, computational model that closely predicts human performance on a range of irregular arrays, albeit with a tendency to underestimate the poorest human solutions. Finally, this hypothesis is intuitively plausible and seems to be echoed in introspective reports by many observers. MacGregor, Ormerod, and Chronicle are to be congratulated, not only on being among the first to pioneer the investigation of a psychological problem of fundamental importance—one that promises to illuminate the nature of intelligence and the relationships between perception and cognition—but also for having formulated a plausible, relatively simple, and testable computational model that is consonant with a range of experimental findings. Nevertheless, this is a new field of study, with some unexpected and intriguing subtleties, and there are several reasons for treating the above attractive conclusions with caution. Indeed, there are some reasons for exploring the opposing view that human performance on TSP tasks is a predominantly local-to-global process. These are articulated in the following section. Difficulties with the Convex Hull Hypothesis Individual Differences First and most generally, the global-to-local hypothesis implies that the initial stage in solving visually presented TSPs is one of an automatic, ‘natural’ perceptual organization, in which the convex hull emerges rapidly as a ‘good figure’ in accordance with the Gestalt principle of Pr¨agnanz (Attneave 1982; Palmer 1999). Indirect evidence in favor of this view, cited by MacGregor and Ormerod (1996) and by MacGregor et al. (2000), includes the uniformly high level of human performance observed in their experiments, the absence of individual differences and an average rank order correlation of individual solutions across different problems that was virtually zero. In agreement with the view that the perception of minimal structure may indeed be a natural tendency of the visual system, Vickers et al. (2001) found that there was a significant and substantial overlap between the tours produced by one group (O), under standard TSP instructions to produce the shortest pathway, and those generated by a different group (G), instructed to draw a circuit that looked most natural, attractive, or aesthetically pleasing. However, Vickers et al. also found that, for both groups, there was a highly significant degree of concordance among observers with respect to the pathways generated for different arrays. These reliable individual differences were confirmed in a second TSP experiment, in which there was a significant correlation between path lengths for individual observers and their scores on Raven’s Advanced Progressive Matrices, a test of general intelligence (APM: Raven, Court, & Raven 1988). Both the existence of reliable individual differences in TSP performance and their significant correlation with intelligence have also been confirmed by Vickers, Mayo, Butavicius and Hughes (submitted), in a study of performance by 69 observers on five different random arrays, each with 50 points. In this case, APM scores for individual observers correlated significantly with their path lengths for each of the five arrays, considered separately, as well as overall. Yet further confirmation of reliable individual differences and of their association with intelligence has been obtained by Vickers, Heitman, and Hughes (submitted). These authors found that individual observers’ path lengths for two 50-point random TSP arrays correlated significantly with APM scores and also with their performance on two different optimization problems, a Minimum Spanning Tree and a Generalized Steiner Tree

TSP Performance 7 problem (e.g., Balakrishnan 1991; Hwang, Richards, & Winter 1992). Taken together, these results indicate that both the general and the more specific formulations of the global-to-local hypothesis need at least to be modified. When arrays are less sparsely populated than most of those employed by MacGregor, Ormerod, and Chronicle, and when the distribution of nodes is random, rather than irregular but constrained, then consistent individual differences do begin to emerge and these are reliably related to cognitive ability. This suggests that TSP performance is not guided by an automatic, instantaneous, global perceptual organizing process, made possible by a massive parallel processing that is uniform for the species. Instead, it suggests that, if performance is guided by global properties of the array, then these global properties are themselves extracted by means of local, intelligently guided processes. Such a view would be consistent with the finding that the achievement of the ‘global’ organization of figure-ground differentiation takes an appreciable time that is systematically related to stimulus parameters (such as relative area), and that individual differences in differentiation times show evidence of a tradeoff between speed and accuracy, and are highly correlated with the times taken in a more cognitive task involving judgments of relative frequency (Vickers 1979, pp. 321-324). Artificial Stimuli Lee and Vickers (2000) point to a second and more consequential set of restrictions on the generality of the conclusions reached by MacGregor, Ormerod, and Chronicle—the nature of which also explains the apparent discrepancies between their findings and ours with respect to individual differences. The convex hulls for the 10and the 20-point arrays employed by MacGregor and Ormerod (1996) were constructed by evenly spacing a number of boundary points on a circle and then displacing each point along the circle by a small random angle (±5◦ ). The nonboundary points were then distributed randomly within a circular or toroidal area within the initial circle. As MacGregor and Ormerod acknowledge, the circular form of the problems might “implicitly have prompted subjects to pay attention to the boundary of the convex hull” (1996, p. 537). The likelihood of this is increased by the fact that the probability that a random configuration of nonboundary points would fall entirely within the prescribed areas is negligibly small in most cases. Its likelihood is increased still further by Lee and Vickers’ (2000) demonstration that the probability of having a large number of points on the convex hull of a 10- or 20-point array is also extremely small. In the view of Lee and Vickers (2000), these combined constraints were likely to have increased the saliency of the convex hull for observers, thereby artifactually biasing the results in favor of MacGregor and Ormerod’s hypothesis. In their reply to this criticism, MacGregor and Ormerod (2000, p. 1052) point out that MacGregor and Ormerod (1996) also reanalyzed previously unpublished data from Lee (1985) for 13 TSPs, ranging from 10 to 60 nodes, that had been randomly generated, and that “the pattern of results was consistent with those obtained with [the] constrained TSPs”. However, what MacGregor and Ormerod (1996) actually reported was that, for Lee’s data, there was a correlation of r = .84 between the number of interior points and the percentage over the optimal for the human solutions. There are two problems with this. The first is that, if these arrays were indeed randomly generated, then the number of interior points will be confounded with the total number of points in each array. The finding that performance declines as a function of the total number of points in a (randomly constructed) array is unsurprising and would

TSP Performance 8 be predicted by most, if not all, models of human performance on TSPs (as, indeed it is by most computational procedures). Second, as implied by MacGregor and Ormerod (1996), and as confirmed by subsequent analysis by the present authors, there was no significant correlation between the number of interior points and the mean percentage by which total path length exceeded the optimal in either of their own two experiments. Thus the two patterns of results may be consistent (if we assume that MacGregor and Ormerod’s results were constrained by ceiling effects, but that those of Lee were not). However, the two patterns are certainly not identical. MacGregor and Ormerod (2000, p. 1052) go on to cite a further experiment by MacGregor et al. (1999), employing two 48-point problems, which, they claim, confirmed that “boundary points guide the solution process”. However, beyond the fact that observers produced solution lengths that were, on average, closer to the optimal for the regular than for the random array, these authors report no analysis that is relevant to the convex hull hypothesis. It is therefore not clear what evidence MacGregor and Ormerod (2000) have in mind in citing this experiment as supporting their hypothesis. Model Selection Issues This difficulty draws attention to a further problem in evaluating the convex hull hypothesis. This is the problem that there are several different response measures that might be interpreted as evidence of the effects of following a solution procedure based on the convex hull. These include: the number of different solutions produced by observers; the number of observers finding an optimal solution; mean total path length (standardized in terms of some random sample of solutions); the mean percentage by which observers’ solutions exceeded the estimated optimal path length; and an information-theoretic measure of response uncertainty, based on the probabilities (over participants) of each possible pairwise connection between the points in an array. In addition, MacGregor and Ormerod (1996) cite the number of indentations and the proportion of observers’ solutions that sequentially connect the points on the convex hull. One problem here is that these separate measures may or may not be correlated, depending on the solution process that is envisaged. For example many algorithms for solving TSPs, such as the CHCI algorithm described by Golden, Bodin, Doyle and Stewart (1980) will produce only one solution to a given problem, even for different choices of a starting node. For algorithms with this property, response uncertainty will always remain constant, independent of the amount by which solutions exceed known optima. In other words, it is not possible to determine the extent to which different response measures may be treated as equivalent, or used to replace each other, or interpreted as independent sources of evidence, without reference to some specific hypothesized solution process. It might be argued (and we would agree) that there is at least one response measure that is of primary importance, namely, the (standardized) amount by which a solution exceeds a known optimum length. After all, the observer is explicitly instructed to reduce this amount to a minimum (subject to constraints), and this is the single goal of the task. The question then arises as to the relationship between this measure and other indices of performance under alternative hypotheses about the solution procedure followed by observers. For example, MacGregor and Ormerod (1996, p. 533) and MacGregor et al. (2000, p. 1184) cite the finding that a very high proportion of their observers “adhered to [the] principle” of connecting the points on the convex hull in order, and interpret this as “consistent with the hypothesis” that observers follow this as a procedure for arriving at a solution. Unfortunately, the observers produced optimal or near-optimal

TSP Performance 9 performance on all problems. Thus, the finding that observers tended to follow the points on the convex hull sequentially is tantamount to an empirical verification of the general principle that the optimal path for every Euclidean TSP will connect the convex hull points in order. Exactly the same result is produced by a number of algorithms, such as the elastic net (Durbin & Willshaw 1987) or simulated annealing routines (Reinelt 1994), which consistently produce optimal or near-optimal solutions. However, these algorithms incorporate very different processes, both from each other and from algorithms based on the convex hull. To the extent that any solution procedure (capable of providing a range of solutions) produces optimal or near-optimal solutions, and inevitably connects the convex hull vertices in order, that procedure will also tend to produce a reduced measure of response uncertainty when the number of points on the convex hull is increased. The reason for this is simply that, when more points fall on the convex hull, more points are constrained to be connected in that order in any optimal solution, and there can only be a reduced response uncertainty in consequence. In other words, neither the proportion of solutions following the convex hull in sequence nor the response uncertainty measure are influenced by the number of nonboundary points in ways that are unique to the global-to-local convex hull hypothesis of MacGregor, Ormerod, and Chronicle. These last difficulties are an instructive illustration of a more profound but widely existing problem of model identifiability, discussed by Roberts and Pashler (2000), among others. This is the desirability of designing experiments, in which different hypotheses make contradictory predictions, thereby permitting “strong inference” (Platt 1964). When the same data can be predicted by a number of alternative hypotheses of comparable complexity, but incorporating different assumptions, then the data cannot be interpreted as providing support for any one hypothesis in preference to the alternatives (Myung, Balasubramanian, & Pitt 2000). This general problem is most directly evident in the relationship between intersections and convex hull points. Quintas and Supnick (1965) provide proofs of the two “folk theorems” enunciated by Flood (1956): (1) that, in the Euclidean plane, the optimal tour cannot intersect itself, and (2) if a tour does not visit the vertices of the convex hull in order, it is not optimal. Their proof of the second theorem (Quintas & Supnick 1965, p. 979) is essentially that a tour not following the convex hull has an intersection, and so, by the result established in the first theorem, cannot be optimal. In this way, Quintas and Supnick (1965) show that not following the convex hull in a tour implies that the tour has an intersection. By negation of the consequent, it is clear that, if a tour does not intersect itself, then it must follow the convex hull. This means that the avoidance of intersections is not a further independent piece of evidence that supports the convex hull hypothesis of MacGregor, Ormerod, and Chronicle. Instead, the fact that observers generally followed the convex hull vertices in sequence could logically follow from their avoidance of intersections.

Local-to-Global Processing In view of the above strictures, it is clearly desirable to formulate alternative approaches to understanding human performance on visually presented TSPs. One approach that seems quite different to that of MacGregor, Ormerod, and Chronicle is to suppose that observers are primarily guided by local processing constraints. In the field of computer

TSP Performance 10 science, the prototype for this kind of approach is the nearest neighbor (NN) algorithm (Golden et al. 1980). The NN algorithm begins at a node (selected at random) and produces a tour by sequentially linking the nearest point to the last previously visited point. In the case of open tours, this procedure is repeated until all the nodes have been visited. In the case of closed tours, once all nodes have been visited, a link is made from the node last visited to the original, starting node. Despite its simplicity, the NN algorithm has a number of attractive features as a model of human TSP performance, as discussed in some detail by both MacGregor et al. (2000) and Graham et al. (2000). First, although the visual perception of an overall structure (such as the convex hull) may be achieved rapidly by means of parallel processing, the actual production of a TSP solution takes place in a serial, link-by-link order, similar to the operation of the NN algorithm. Second, although many algorithms provide only one solution to a given problem, the NN algorithm produces a range of different solutions (depending upon the selection of starting node). In this, it resembles the performance of a group of human observers, who tend to produce a variety of solutions, particularly when problems become more difficult. (We are not aware of any evidence on how consistently an individual observer might solve a given problem, but we surmise that observers would not consistently produce the same solution to a single problem presented on different occasions among a series of different problems.) At the same time, few (if any) researchers would seriously suggest that observers follow a simple NN procedure. MacGregor and Ormerod (1996) found that the mean human solutions (averaged across observers and across problems) in both their experiments were consistently superior to the mean solutions produced by the NN algorithm. Similarly, Graham et al. (2000) found that, for smaller problems (with 6 or 10 nodes), the solutions produced by the NN algorithm were comparable to those of human observers, but that, when the problems were larger (20 or 30 nodes), the NN algorithm produced substantially longer tours. If these results had been the reverse, it would be reasonable to attempt to explain them by supposing that human performance was also limited by some other factor, such as the accuracy with which inter-node distances can be judged. However, the fact that the superior performance by the human observers lies beyond the range of results predictable by the NN algorithm means that the algorithm is seriously deficient. Despite its deficiency, it is not difficult to think of ways in which the NN algorithm might be improved. For example, one possibility would be to combine the NN algorithm with the heuristic of avoiding self-intersecting paths. There appears to be overwhelming evidence that observers’ solutions contain very few crossed paths. The problem is how to test whether this is an intentional strategy or an incidental result of adhering to some other procedure. In addition to the above severe strictures on the evidence cited in favor of the convex hull hypothesis, Lee and Vickers (2000) draw attention to the constraint that, when TSP configurations are constructed randomly, then the variation in the number of nodes that fall on the convex hull is relatively small and both the variance and the mean number of such nodes exhibit asymptotic behavior. As a result, as configurations of increasing size are considered, the concept “number of interior or nonboundary nodes” becomes equivalent to “total number of nodes”. Of course, as MacGregor and Ormerod correctly state in their reply, “it is possible to construct problems of any size in which the total number of points and the number of nonboundary points vary orthogonally For example, this could be done by selecting problems from the upper and lower ranges of the distributions of

TSP Performance 11 nonboundary points shown in Lee and Vickers’s Figure 1” (MacGregor & Ormerod 2000, p. 1052).

Experiment In view of the controversy concerning the evidence relevant to the convex hull hypothesis, and in view of the plausible opposing hypothesis that many results could be attributed to a tendency by observers simply to avoid self-intersecting pathways, an experiment was carried out closely conforming to the above suggestion by MacGregor and Ormerod (2000). Participants There were 20 participants, the majority of whom (19 out of 20) were either studying or had studied at a tertiary level. Their ages ranged from 17 to 40 years, with a mean age of 23. Stimuli The test arrays were drawn from a pool of 20,000 randomly generated 40-node arrays, and were selected according to the number of nodes they had on the convex hull and their number of intersections, as calculated by counting the number of intersections over all possible pairs of line segments. Figure 1 shows each of the 20,000 randomly generated arrays in terms of their number of nodes on the convex hull and number of intersections, together with the marginal distributions for these two measures. Using this information, we defined 9 problem types, following a 3 (high, medium and low number of points on the convex hull) × 3 (high, medium and low number of intersections) factorial design. The mean number of intersections crossings was 63,542, and this was designated as the midpoint, with high (66,341 intersections) and low (60,743 intersections) selected to be 1.5 standard deviations (2,799 intersections) above and below the mean respectively (tolerance for selection was around ±5% of the standard deviation, or ±93 intersections). The mean number of nodes on the convex hull was calculated in a similar manner (Mean = 9.61, S.D. = 1.587). The medium number of nodes on the convex hull was selected to be either 9 or 10 points (alternated randomly), with a low number of nodes as 7, and a high number as 12. The locations of the 9 problem types, showing the 3 × 3 grid, are displayed as white markers (or a pair of white markers, in the case of medium convex hull points) in Figure 1. Eight arrays from the pool of 20,000 that met the criteria were chosen at random for each of the 9 problem types. Procedure Observers attended two test sessions. Each test session ran for approximately two and a half hours. The participants were given an hour to complete 18 problems. After a halfhour break, the observers completed another 18 problems. Each test session contained two of each of the 9 types of problem, and, within each session, the arrays were presented to observers in a quasi-random manner.

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Number of Intersections Figure 1: The selection of experimental stimuli. The black points represent 20,000 randomly generated arrays, characterized in terms of the number of points on the convex hull and the number of intersections. The histograms show the marginal distributions for these two measures. The 3 × 3 grid of white markers indicate the low, medium and high convex hull and intersection values used to select experimental stimuli.

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Convex Hull Points (H) Figure 2: The mean deviation from optimality, taken across all participants solutions to all 8 problems, for each combination of points on the convex hull and number of intersections.

The arrays were presented on computer color monitors. In each array, the nodes were distributed within a square field 15 × 15 cm. Observers were instructed to draw the shortest pathway through all of the nodes in each array, passing through each node only once and returning to the starting node. They were allowed to delete and re-draw any of the connections, and could begin their pathways at any node in the array. Immediately prior to commencing each test session, observers were presented with two examples of possible solutions to a TSP, one of which contained an inter-point crossing. After each problem was completed, observers were presented with the optimal solution and a score of the percentage difference between their own solution and the optimal solution.

Results Deviations from optimality Optimal solutions to the test arrays were calculated by means of a simulated annealing heuristic (Reinelt 1994; Press 1992). To permit comparison across the different problem types, observers’ pathway lengths were expressed as percentage above the optimal score. Thus, an observer’s score for a perfect pathway would be zero, and any pathway that exceeds the optimum will have a positive score.

TSP Performance 14 Figure 2 shows the change in the deviation from optimality measure, averaged across all 8 arrays for each problem type, in terms of the convex hull and intersection factors. Surprisingly, and contrary to both the findings and the convex hull hypothesis of MacGregor and Ormerod (1996), mean scores increased as a direct function of the number of nodes on the convex hull. Conversely, mean scores decreased as the number of intersections increased. A univariate analysis of variance was carried out, with the deviation from optimality as the dependent variable, and the number of intersections and the number of points on the convex hull as independent variables. Levene’s test indicated heterogeneity of variance (F [8, 1431] = 3.44; p < .001). A significant difference in mean deviation from optimality scores was found for both the number of intersections (F [2, 1439] = 9.042; p < .0001), and the number of points on the convex hull (F [2, 1439] = 9.855; p < .0001). No interaction effect was found between the number of intersections and the number of points on the convex hull (F [4, 1439] = 0.669; p > .05). Due to the heterogeneity of variance in the data set, all post-hoc analyses were carried out using Tamhane’s T2, which does not rely upon equality of variance. Mean deviation from optimality scores for arrays with high numbers of nodes on the convex hull were significantly greater than those for arrays with low (p = .0001) or medium (p = .025) numbers of hull-bound nodes, but no significant difference was found between mean deviation from optimality scores for arrays with medium and low numbers of hull-bound points (p = .24). Mean deviation from optimality scores for arrays with low numbers of intersections were significantly greater than those for arrays with medium (p = .02) or high (p = .0001) numbers of intersections, but no significant difference was found between mean scores for arrays with medium and high numbers of intersections (p = 0.452). Response Uncertainty In addition to deviations from optimality in the case of individual observers, a group measure of response uncertainty was calculated for each array. The frequency with which each point in the array was connected with each other point was counted and an n × n matrix constructed. Probabilities pi were obtained by dividing the frequencies by the number of participants (20), and then applying Shannon’s (1948) standard information theory formula, given in Equation 1, to calculate the total response uncertainty (H), with k being the total number of connections made by the participants. H=

k
i=1

pi log2 pi .

(1)

Figure 3 shows the change in the response uncertainty measure, averaged across all 8 arrays for each problem type, in terms of the convex hull and intersection factors. A univariate analysis of variance was performed, with response uncertainty as the dependent variable, and number of nodes on the convex hull and number of intersections as the independent variables. Levene’s test of equality of error variances was significant (F [8, 63] = 1.165, p = .334) indicating homogeneity. A significant difference in response uncertainty was found for intersections (F [2, 71] = 10.043, p = .0001), but not for number of nodes on the convex hull (F [2, 71] = 0.77, p = .467). No interaction effect was found (F [4, 71] = 1.25, p = .299). Post hoc analysis indicated a significant difference between

TSP Performance 15

Low 30

25

Intersections (X)

Response Uncertainty

35

Med

High Low

Med

High

Convex Hull Points (H) Figure 3: The mean deviation from optimality, taken across all participants solutions to all 8 problems, for each combination of points on the convex hull and number of intersections.

TSP Performance 16 the path uncertainty of arrays with low numbers of intersections and medium (p = .002) and high numbers (p = .0001). No significant difference was found between the response uncertainty measures for medium and high numbers of nodes (p = .297). Having presented these results, we acknowledge our sensitivity to criticisms of NullHypothesis Significance Testing (NHST) as a method of scientific inference (e.g., Cohen 1994; Edwards, Lindman, & Savage 1963; Howson & Urbach 1993; Hunter 1997; Lindley 1972). In particular, we acknowledge that NHST violates the likelihood principle, and so does not satisfy a basic requirement for rational, consistent and coherent statistical decision making (Lindley 1972). Accordingly, the appendix provides a separate analysis that uses the Bayesian approach to statistical inference. Fortunately, these analyses point towards to the same theoretical conclusions as the NHST analyses.

Discussion Both the NHST and the Bayesian analyses coincide in supporting the interpretation that both the number of points on the convex hull and the number of intersections have independent effects on deviations from optimality. The direct relation between the number of hull-bound points and the extent of deviations from optimality contradicts the convex hull hypothesis of MacGregor and Ormerod (1996) and is also at variance with the absence of any relationship between these variables in the experiment of Vickers et al. (2001). In contrast, the inverse relation between number of intersections and deviation from optimality is consistent with a local-to-global processing procedure, in which observers avoid constructing pathways that intersect, either as a deliberate strategy or as a concomitant of some more fundamental process. The NHST and Bayesian analyses also agree in indicating that number of intersections has an inverse effect on path uncertainty, and that there is no evidence that number of hullbound points has an effect or that these two factors interact. This outcome fails to support MacGregor and Ormerod’s interpretation of their finding of an inverse relation between number of hull-bound points and path uncertainty. Again, however, it is consistent with the hypothesis that observers are effectively avoiding paths that cross. Taken in conjunction with the interpretative weaknesses in evidence cited in favor of the global-to-local convex hull hypothesis, these conclusions suggest that there is no evidence that can be adduced as unqualified support for the convex hull hypothesis and there appears to be some clear evidence that runs counter to it. In contrast, the view that observers avoid intersections is consistent with the findings from several experiments, has additional explanatory power, and is supported by the specific manipulations carried out in the present experiment. There appears to be no unequivocal, obvious explanation for the surprising direct relation between hull-bound points and deviation from optimality. As may be judged from Figure 1, the configurations selected as problems are quite representative of the population of randomly constructed 40-node arrays. It seems unlikely that an opposite outcome could be obtained with another method of selecting configurations that would retain orthogonality in the values of the two independent variables. However, it remains possible that the configurations selected are characterized by greater difficulty associated with the way in which the nodes are distributed. An answer to this question may well depend on the development of further measures of problem difficulty, similar to the number of potential intersections in a TSP stimulus array.

TSP Performance 17 Meanwhile, the possibility that much of TSP performance might be explained by the locally focused tendency of observers to avoid crossings seems worth pursuing further. It is known that observers rarely, if ever, see paths of apparent motion that intersect (Kolers 1972, p. 77). One possible reason is that such paths do not constitute least distances. Within a random configuration of nodes, each node has a distance to every other node, and among these there is a least distance. Thus, across all nodes, there is a set of least distances. Vickers (2001) and Vickers and Preiss (submitted) have recently presented a considerable amount of evidence consistent with a generative transformational model of visual perception, according to which the spatial structure within a static array and the spatio-temporal structure of motion between two successive arrays are both determined by the distributions of least distances between array elements. It seems possible that least distances (which never cross) within a random array might be an important ingredient in human performance on visually presented TSP tasks. Such a view would be consistent with MacGregor and Ormerod’s argument that the task of solving TSPs coincides with natural tendencies of the visual system. However, it implies that the explanation of both perceptual organization and of TSP performance is to be found in local-to-global processing, rather than the converse.

References Attneave, F. (1982). Pr¨agnanz and soap bubble systems: A theoretical exploration. In J. Beck (Ed.), Organization and Representation in Perception, pp. 11–29. Hillsdale, NJ: Erlbaum. Balakrishnan, V. K. (1991). Introductory Discrete Mathematics. New York: Dover. Carlin, B. P., & Louis, T. A. (2000). Bayes and Empirical Bayes Methods for Data Analysis (Second ed.). New York: Chapman & Hall. Cohen, J. (1994). The earth is round (p < .05). American Psychologist 49, 997–1003. Dantzig, G. B., Fulkerson, D. R., & Johnson, S. M. (1959). On a linear-programming combinatorial approach to the travelling salesman problem. Operations Research 7, 58–66. Durbin, R., & Willshaw, D. (1987). An analogue approach to the travelling salesman problem using an elastic net method. Nature 326, 689–691. Edwards, W., Lindman, H., & Savage, L. J. (1963). Bayesian statistical inference for psychological research. Psychological Review 70 (3), 193–242. Flood, M. M. (1956). The travelling salesman problem. Operations Research 4, 61–75. Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall. Golden, B., Bodin, L., Doyle, T., & Stewart, W. (1980). Approximate travelling salesman algorithms. Operations Research 28, 694–711. Graham, S. M., Joshi, A., & Pizlo, Z. (2000). The traveling salesman problem: A hierarchical model. Memory & Cognition 28, 1191–1204. Howson, C., & Urbach, P. (1993). Scientific Reasoning: The Bayesian Approach. La Salle, IL: Open Court Press.

TSP Performance 18 Hunter, J. E. (1997). Needed: A ban on the significance test. Psychological Science 8 (1), 3–7. Hwang, F. K., Richards, D. S., & Winter, P. (1992). The Steiner Tree Problem. Amsterdam: Elsevier. Kass, R. E. ,& Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association 90 (430), 773–795. Kolers, P. A. (1972). Aspects of Motion Perception. London: Pergamon Press. Krolak, P., Felts, W., & Marble, G. (1971). A man-machine approach toward solving the traveling salesman problem. Communications of the ACM 14, 327–334. Lawler, E. L., Lenstra, J. K., Rinooy Kan, A. H. G., & Schmoys, D. B. (1985). The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Chichester, UK: Wiley. Lee, M. D., & Vickers, D. (2000). The importance of the convex hull for human performance on the traveling salesman problem: A comment upon MacGregor and Ormerod (1996). Perception & Psychophysics 62, 226–228. Lee, R. K. L. (1985). A heuristic approach to the traveling salesman problem. [Unpublished Management Report]. University of Victoria, School of Public Administration. Leonard, T., & Hsu, J. S. J. (1999). Bayesian Methods: An Analysis for Statisticians and Interdisciplinary Researchers. New York: Cambridge University Press. Lindley, D. V. (1972). Bayesian Statistics: A Review. Philadelphia, PA: Society for Industrial and Applied Mathematics. MacGregor, J. N., & Ormerod, T. C. (1996). Human performance on the traveling salesman problem. Perception & Psychophysics 58, 527–539. MacGregor, J. N., & Ormerod, T. C. (2000). Evaluating the importance of the convex hull in solving the Euclidean version of the traveling salesperson problem: Reply to Lee and Vickers (2000). Perception & Psychophysics 62, 1501–1503. MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (1999). Spatial and contextual factors in the traveling salesperson problem. Perception 28, 1417–1428. MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (2000). A model of human performance on the traveling salesperson problem. Memory & Cognition 28, 1183– 1190. Myung, I. J., Balasubramanian, V., & Pitt, M. A. (2000). Counting probability distributions: Differential geometry and model selection. Proceedings of the National Academy of Sciences 97, 11170–11175. Norback, J. P., & Love, R. F. (1977). Geometric approaches to solving the traveling salesman problem. Management Science 23, 1208–1223. Ormerod, T. C., & Chronicle, E. P. (1999). Global perceptual processing in problem solving: The case of the traveling salesperson. Perception & Psychophysics 61, 1227– 1238. Palmer, S. E. (1999). Vision Science: Photons to Phenomenology. Cambridge, MA: MIT Press.

TSP Performance 19 Platt, J. R. (1964). Strong inference. Science 146, 347–353. Polivanova, N. I. (1974). [functional and structural aspects of the visual components of intuition in problem solving]. Voprosy Psikhologii 4, 41–51. Pomerantz, J. R. (1981). Perceptual organization and information processing. In M. Kubovy & J. R. Pomerantz (Eds.), Perceptual Organization, pp. 141–180. Hillsdale, NJ: Erlbaum. Press, W. H. (1992). Numerical Recipes in FORTRAN: The Art of Scientific Computing (Second ed.). Cambridge, UK:: Cambridge University Press. Quintas, L. V., & Supnick, F. (1965). On some properties of shortest Hamiltonian circuits. American Mathematical Monthly 72, 977–980. Raven, J. C., Court, J. H., & Raven, J. (1988). Manual for Raven’s Progressive Matrices and Vocabulary Scales. Section 4: Advanced Progressive Matrices. London: Lewis. Reinelt, G. (1994). The traveling salesman: Computational solutions for TSP applications. Lecture Notes in Computer Science, 840. Berlin: Springer-Verlag. Roberts, S., & Pashler, H. (2000). How persuasive is a good fit? A comment on theory testing. Psychological Review 107 (2), 358–367. Sanocki, T. (1991). Effects of early common features on form perception. Perception & Psychophysics 50, 490–497. Sanocki, T. (1993). Time course of object identification: Evidence for a global-to-local contingency. Journal of Experimental Psychology: Human Perception and Performance 19, 878–898. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics 6 (2), 461–464. Shannon, C. E. (1948). The mathematical theory of communication. Bell Systems Technical Journal 27, 379–243, 623–656. Sivia, D. S. (1996). Data Analysis: A Bayesian Tutorial. Oxford: Clarendon Press. Vickers, D. (1979). Decision Processes in Visual Perception. New York, NY: Academic Press. Vickers, D. (2001). Towards a generative transformational approach to visual perception. Behavioral and Brain Sciences 24, 707–708. Vickers, D., Butavicius, M. A., Lee, M. D., & Medvedev, A. (2001). Human performance on visually presented traveling salesman problems. Psychological Research 65, 34– 45. Vickers, D., Heitman, M., & Hughes, P. (submitted). Optimization and intelligence: Individual differences in performance on visually presented optimization problems. Manuscript submitted for publication. Vickers, D., Mayo, T., Butavicius, M. A., & Hughes, P. (submitted). Optimization and intelligence: Individual differences in performance on visually presented travelling salesperson problems. Manuscript submitted for publication. Vickers, D., & Preiss, K. (submitted). Impressions of form and motion from Glass patterns. Manuscript submitted for publication.

TSP Performance 20 Wilf, H. S. (1986). Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall.

Appendix: Bayesian Analysis The basic idea behind Bayesian statistical inference (e.g., Carlin & Louis 2000; Edwards, Lindman, & Savage 1963; Kass & Raftery 1995; Gelman, Carlin, Stern, & Rubin 1995; Leonard & Hsu 1999; Lindley 1972; Sivia 1996) is to treat the data gathered in experiments as empirical evidence for or against different competing theories. By specifying these theories as quantitative models, it is possible to calculate Bayes Factors, which measure the relative evidence provided by data. A Bayes Factor takes the form of an odds ratio, specifying how much more likely one model is than another, on the basis of the data. Because they are expressed as odds, Bayes Factors have inherent meaning: A Bayes Factor of 10, for example, means that one model is 10 times more likely than the other. Whether such a difference should be regarded as ‘significant’ is a question about what standards of scientific evidence are required for the research problem at hand. To make inferences about the influence the convex hull and intersection factors have on the human performance, we considered six possible theoretical accounts, each of which can be specified as a quantitative model. The first model, called FULL, assumes that both factors affect performance, and that they interact in a completely unconstrained way. This model requires that each of the 3 × 3 = 9 means uses a separate parameter to model performance. At the other end of the spectrum, the second model, called CONSTANT, assumes that neither factor has any effect on performance, and so all of the means can be modeled by the same single parameter. The third model, INTERSECT, assumes that only changes in the number of intersections affect performance, while the fourth model, HULL, assumes that only changes in the number of points on the convex hull affect performance. Both of these models require three parameters, one for each level of the factor that is assumed to affect performance. The fifth model, ONE-WAY, assumes that both factors affect performance in an entirely independent way, so that the change in one does not depend upon the value of the other. This model involves four parameters: two to describe the rates of change for each factor, and two to describe the differences in their initial values. The sixth and final model, TWO-WAY, extends ONE-WAY by assuming that there is some limited dependency. In particular, it assumes that each change in a factor affects performance by a constant amount, but that the value of this constant depends upon the level of the other factors. The TWO-WAY model is more constrained than the FULL model, requiring six parameters: three to describe the rate of change for each of the two factors. The top panel of Figure 4 shows the empirical data, giving the mean deviation from optimality for each possible combination of points on the convex hull and number of intersections This acts as a reference for the bottom six panels, which show the best fits to these data under each of the six models. From these data fits, and the known parametric complexity of the models, it is possible to calculate the Bayesian Information Criterion (BIC: Schwarz 1978) for each model. The BIC values, in turn, allow Bayes Factors between each pair of models to be estimated (e.g., Kass & Raftery 1995). Table 1 summarizes these results, showing the log-likelihood measure of data fit, the number of parameters, the BIC, and the Bayes Factors for each model. The Bayes Factor is taken with respect to the most likely model (i.e., the one with the minimum BIC value), which is the ONE-WAY model.

TSP Performance 21

0.05

Intersections (X)

Deviation from Optimal

0.06

Low Med

High

0.04 Low

Med

High

Convex Hull Points (H) FULL

CONSTANT

INTERSECT

HULL

ONE−WAY

TWO−WAY

Figure 4: The top panel shows the mean deviation from optimality, taken across all participants solutions to all 8 problems, for each combination of points on the convex hull and number of self intersections. The bottom six panels shows the best fits to these data using 6 different models, as discussed in the text.

TSP Performance 22

Table 1: The log-likelihood data fit, parametric complexity, BIC value, and Bayes Factor for each of the 6 models in relation to the deviation from optimality score. The Bayes Factors are taken in relation to the most likely model, which is the ONE-WAY model. Model Name

Log-Likelihood Parametric Data Fit Complexity

BIC Value

Bayes Factor

FULL

10.55

9

-1.32

26.63

CONSTANT

4.96

1

-7.72

6.55

INTERSECT

7.27

3

-7.95

5.43

HULL

7.65

3

-8.70

3.97

ONE-WAY

10.12

4

-11.46

1.00

TWO-WAY

10.18

6

-7.19

7.86

What the Bayesian analysis shows is that the data provide the most evidence for the ONE-WAY model, with the HULL model being 3.97 times less likely, the INTERSECT model being 5.43 times less likely, and so on. It is worth noting that the data provides the least evidence for those models that assume some form of interaction between the two factors. Our conclusion, on the basis of these findings, is that both the number of points on the convex hull, and the number of intersections influence performance, but do so entirely independently of each other. This conclusion is consistent with the one achieved using NHST methods, which found two significant one-way effects, but no significant interaction. The top panel of Figure 5 shows the pattern of change in the response uncertainty measure for each possible combination of points on the convex hull and number of intersections. Once again, the bottom six panels of Figure 5 shows the best fits to these data under each of the six models. Table 2 gives the various Bayesian measures for each model against response uncertainty. In this case, the INTERSECT model has the minimum BIC value, and so is the basis for the Bayes Factors. Our conclusion from this analysis is that increasing the number of intersections clearly does decrease response uncertainty, but there is no clear evidence that the number of points of the convex hull has an effect. Once again, this conclusion is consistent with NHST methods, which found a significant one-way effect for intersections, but none for the convex hull, and no significant interaction. There are two qualifications that should be attached to these conclusions. The first is that our findings relate only to the range of values for the two factors considered in the experiment. By construction, these ranges cover the situations that occur in naturally occurring (i.e., randomly generated arrays), but it remains possible that artificially generated arrays might give different results. The second qualification is that the models we have considered, while covering a variety of the most interesting theoretical alternatives, are by no means exhaustive. It remains possible that other models could be constructed that are more likely than the ONE-WAY and INTERSECT models and provide a different

TSP Performance 23

Low 30

25

Intersections (X)

Response Uncertainty

35

Med

High Low

Med

High

Convex Hull Points (H) FULL

CONSTANT

INTERSECT

HULL

ONE−WAY

TWO−WAY

Figure 5: The top panel shows the mean response uncertainty, taken across all 8 problems, for each combination of points on the convex hull and number of self intersections. The bottom six panels shows the best fits to these data using 6 different models, as discussed in the text.

TSP Performance 24

Table 2: The log-likelihood data fit, parametric complexity, BIC value, and Bayes Factor for each of the 6 models in relation to the response uncertainty measure. The Bayes Factors are taken in relation to the most likely model, which is the INTERSECT model. Model Name

Log-Likelihood Parametric Data Fit Complexity

BIC Value

Bayes Factor

FULL

-1.70

9

23.17

19.17

CONSTANT

-6.97

1

16.14

5.33

INTERSECT

-2.82

3

12.23

1.00

HULL

-6.02

3

18.62

9.44

ONE-WAY

-2.64

4

14.07

2.51

TWO-WAY

-2.55

6

18.29

8.27

interpretation of the data. We did experiment with models beyond the 6 reported here for both deviation from optimality and response uncertainty, and failed to find any superior account, but cannot completely discount the possibility that one exists.