OPTIMIZATION AND INTELLIGENCE: INDIVIDUAL DIFFERENCES IN PERFORMANCE ON THREE TYPES OF VISUALLY PRESENTED OPTIMIZATION PROBLEMS

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1 OPTIMIZATION AND INTELLIGENCE: INDIVIDUAL DIFFERENCES IN PERFORMANCE ON THREE TYPES OF VISUALLY PRESENTED OPTIMIZATION PROBLEMS Douglas VICKERS, Megan HEITMAN, Michael D. LEE, and Peter HUGHES University of Adelaide, South Australia 5005 Although problem solving is an essential expression of intelligence, both experimental and differential psychology have neglected an important class of problems, for which it is difficult or impossible for systematic procedures to provide a definitive solution. A study is described, in which participants solutions to three such computationally difficult problems (a Travelling Salesperson, a Minimal Spanning Tree, and a Generalized Steiner Tree problem) all showed consistent individual differences that intercorrelated reliably and correlated moderately with scores on Raven s Advanced Progressive Matrices. The results are interpreted in terms of a theory of visual perception based on detecting minimal structures and selecting transformations that maximize self-similarity. This interpretation is discussed in the context of a perspective in which intelligence is viewed as the capacity to solve problems that (either in principle or in practice) defy definitive solution by a mechanical algorithm. 1. Introduction The ability to solve problems is a central cognitive activity and is considered an essential ingredient in any well conceived notion of intelligence (Resnick & Glaser, 1976; Sternberg, 1982). Despite the importance of problem-solving, however, there exists no comprehensive taxonomy of problems and many problems chosen for study have only weak claims to being ecologically representative. For example, one common feature of most problem-solving tasks is the possession of well defined initial and end states that must be linked by a series of operations. Such problems typically have a unique answer that can be arrived at by an algorithmic procedure (i.e., a systematic, finite series of steps). Indeed, solvability in terms of a unique solution has earlier been suggested as a desirable criterion for any problem-solving task (Ray, 1955). On the other hand, there is an important class of real-world problems that have been neglected in both experimental and differential psychology. These are problems of combinatorial optimization for which there is no algorithm that can be guaranteed to produce a definitive solution within a time that is practicable. A classic example is the Travelling Salesperson Problem (Lawler, Lenstra, Rinooy Kan, & Shmoys, 1985), historically referred to as the Travelling Salesman Problem (TSP). The planar Euclidean version of the TSP can be formulated as follows: Given a set of n interconnected towns, devise an itinerary that visits each town exactly once, returns to the starting point, and ensures that the total distance travelled is as short as possible. To arrive at a definitive solution to this problem entails an exhaustive consideration of

2 (n 1)!/2 pathways. This is feasible when n is a small number, such as 5, and there are only 12 distinct pathways to consider. However, when n is increased to just 25, the number of pathways becomes so immense that a computer evaluating a million possibilities a second would take almost 10 billion years to evaluate them all (Stein, 1989). Despite these computational difficulties, one striking finding is that human performance on visually presented TSP tasks is impressively close to optimal, and, in many cases, outperforms that of the most powerful computational heuristics, at least when n is less than 100 (Graham, Joshi, & Pizlo, 2000; MacGregor & Ormerod, 1996; MacGregor, Ormerod, & Chronicle, 1999; MacGregor, Ormerod, & Chronicle, 2000; Polivanova, 1974). On this basis, MacGregor and Ormerod (1996) have concluded that the solution processes employed correspond to natural organizing processes of the visual system. This conclusion appears to be consistent with a variety of other findings concerning the spontaneously perceived or aesthetically constructed organisation of the nodes in TSP arrays, as well as the judged goodness of alternative solutions, viewed as abstract patterns (Garner, 1970; Polivanova, 1974; Pomerantz, 1981; Ormerod & Chronicle, 1999; Vickers, Butavicius, Lee, & Medvedev, 2001). In two recent papers, Vickers et al. (2001) and Vickers, Mayo, and Butavicius (submitted) have argued that, in view of the prevalence of optimization processes and their practical, scientific, biological, socio-economic and psychological significance, it is important to investigate human abilities in solving computationally difficult or intractable optimization problems, such as the TSP. For example, such problems arise in visual perception, in realistic problems of memory search and reasoning, in the precise control of movement with a large number of degrees of freedom, and in search and foraging behavior. Indeed, Vickers, Mayo, et al. (submitted) have suggested that human intelligence may have evolved especially to deal with computationally intractable problems, and have gone on to propose that intelligence may be defined as the capacity to find near-optimal solutions to problems for which (in principle or in practice) there can be no mechanical algorithm for arriving at a definitive solution. This perspective on intelligence, they argue, focuses attention on important issues, suggests new ways in which these issues might be investigated, and is capable of reconciling competing views of intelligence. In a preliminary exploration of this perspective, Vickers et al. (2001) examined the ability of 40 university student participants to find near-optimal solutions to a visually presented TSP consisting of 40 randomly distributed nodes. Standardized measures of the extent to which solution path lengths exceeded a known optimal were found to have a modest correlation (Spearman rho = -0.36) with intelligence, as measured by performance on Raven s Advanced Progressive Matrices (APM; Raven, Court, & Raven, 1988). Because of the possibility of ceiling effects in this result, Vickers, Mayo, et al. (submitted) examined the performance of a wider sample of 69 participants on both the APM and on five visually presented 50-node TSP arrays. In this case, the correlation between APM scores and the mean percentage deviation of all TSP solutions from their corresponding optimal values was moderately high (Pearson r = -0.48). The average of the intercorrelations for the individual TSP arrays was also moderately high (Pearson r = 0.53). These results lend some support to the view that intelligence may be related to the need to deal with computationally diddicult or intractable problems as exemplified in the TSP. At the same time, the results also raise a number of questions. For

3 example, MacGregor and Ormerod (1996) have suggested that people s impressive performance on visually presented TSPs might be explained by the coincidental circumstance that such tasks happen to permit the operation of other, independently grounded perceptual principles, such as a global-to-local Gestalt preference for seeing stimuli as convex shapes. On the other hand, Vickers, Lee, Dry, and Hughes (submitted) and Vickers, Bovet, Lee, and Hughes (submitted) have argued that it is the local-to-global perception of some form of minimal structure, through the detection of least distances between nodes, that is a natural tendency of the perceptual (and perhaps the cognitive) system, and that Gestalt principles represent different instantiations of this tendency. The following experiment was designed to explore the notion that intelligence is associated with a more general ability to arrive at near-optimal solutions to computationally intractable problems other than the TSP and to investigate the possibility that such an ability is largely determined by automatic perceptual organizing tendencies or represents an interplay between perceptual organization and cognitive control. To test the generality of the association between intelligence and the detection of minimal structures, and to investigate the possible dependence of such an association on automatic perceptual processes, three contrasting optimizing tasks were selected. The first was a visually presented TSP task, with 50 randomly distributed nodes, as illustrated in Figure 1a. The second was a Minimum Spanning Tree Problem (MSTP), also with 50 nodes, as illustrated in Figure 1b. A minimum spanning tree is the shortest path that directly links all the nodes in an array. Like the TSP, this is a combinatorial problem. However, it has fewer constraints, because the path does not have to be continuous and closed, and a node can be connected by more than two links. Unlike the TSP, the MSTP is computationally tractable (Ahuja, Magnanti, & Orlin, 1993). The optimal solution is an open, branching path system, which directly links all the nodes, and in which branches occur at the nodes. Insert Figure 1 about here The third problem was a 15-node Generalized Steiner Tree Problem (GSTP) (Hwang, Richards, & Winter, 1992). The elementary, three-node Steiner Tree is based on the geometric properties of a triangle, according to which the shortest path connecting three nodes in a plane can be found by determining the so-called Fermat point, and creating a path of links that branch at that point (as illustrated in Figure 2a). If all the angles in a triangle, ABC, are less than 120º, then the Fermat point, P, is the uniquely determined point within the triangle, from where the angles APC, CPB, and BPA are all exactly 120º. However, if one of the angles is at least 120º (say, the angle ACB), then P must coincide with C. Insert Figure 2 about here Generalized Steiner Tree Problems contain more than three nodes, and the solutions to these problems look like combinations of solutions found in three-node Steiner Tree Problems, with additional nodes (P 1, P 2, P k ) as branch points to create

4 the minimum connections between the original nodes, as illustrated in Figure 2b. Given a set of nodes, it is a priori unclear how many further nodes need to be added in order to construct a minimum path. However, there is a proven maximum of (n 2) additional nodes. The TSP task was included as a reference and to replicate previous results. Vickers and Preiss (submitted), Vickers, Mayo, et al. (submitted) and Vickers, Lee et al. (submitted) have argued that an important perceptual component in the TSP solution process is the detection of a structure of least distances between each node and every other node. Vickers, Mayo, et al. (submitted) found that, the higher the number of closest links in subjects solutions, the closer human performance approached the optimum. At the same time, the optimal solution may not include all the least distances. Thus, the TSP makes possible a tension between a spontaneous perceptual tendency and the cognitively mediated demand of constructing a circuit that is shortest overall. On the other hand, the MSTP does include all the least distances, although it may also require additional links to connect up substructures of least distances. Viewed in this way, the solution procedure for the MSTP is arguably more compatible with a locally focussed, spontaneous perceptual process of detecting least distances. Consistent with this argument, Zahn (1971) found that algorithms based on minimum spanning trees gave a good account of the perception of Gestalt clusters and organization in dot arrays. Also in agreement with such a process is the finding by Pomerantz (1981) that subjects, asked to connect up the dots in arrays, to illustrate how they perceived the arrays, frequently connected the dots into the shortest possible (minimum spanning tree) path structure. In contrast, the GSTP task is arguably the least compatible with natural perceptual tendencies, because it does not simply link the visible nodes directly but also requires the imposition of additional nodes and links that are well separated from any structure formed by direct links and least distances between existing nodes. Consistent with the assumption that this task is more cognitively demanding than the TSP or MSTP tasks is the fact that a pilot experiment suggested that a GSTP task with a much smaller number of nodes (15, as opposed to 50) was required to produce a performance level equivalent to that for either of the other two tasks. 2. Method 2.1. Participants. A total of 48 volunteer undergraduate students from the University of Adelaide, aged between 17 and 50 years (M = 26.2; SD = 9.2), participated in the experiment Problems. Participants provided solutions to two visually presented TSP, MSTP and GSTP arrays. The second of each pair of problems was identical to the first, but was rotated through 90º. The TSP and MSTP arrays are shown in Figures 1 and 2, and each consisted of 50 nodes, the coordinates of which were randomly selected from a uniform (rectangular) distribution. The GSTP array is shown in Figure 3, and consisted of 15 nodes, the coordinates of which were also selected randomly from a uniform distribution. For each type of problem, there was also a practice array. In the case of the TSP and the MSTP, this consisted of 15 randomly distributed nodes, and, for the GSTP, this consisted of an array of 7 randomly distributed nodes.

5 2.3. Design. Participants performed the three types of problem in a single session and in an order that was varied according to a latin square design (Fisher & Yates, 1963), so that equal numbers of the 48 participants solved each type of problem in each of the six different possible orders. After solving a practice problem, participants solved the untransformed and transformed versions of that type of problem, before being presented with the practice array for the next type of problem. In addition, participants completed Set 1 (practice) and Set 2 (test) of Raven s APM (Raven, Court, & Raven, 1988) at a separate, prior session Instructions. Participants were given instructions by way of an introductory information sheet, in the form of verbal instructions by the experimenter, and by means of instructions on the computer screen at the start of each session (which could be re-displayed at any time by clicking on a button). There were no time constraints on performance, but participants were asked to complete the tests as quickly and as accurately as they could. For the TSP, participants were asked to connect all the nodes in a continuous tour that visited each node once only, returned to the starting node, and was as short as possible. In the case of the MSTP, participants were asked to connect all the nodes by making a system of links (or paths) between them, using as many links as necessary, but only between existing nodes, and creating a path system with the shortest possible overall length. With the GSTP, participants were asked to connect all the nodes by making a system of links (or paths) with the shortest overall length. In this case, however, participants were told that the existing nodes did not have to be linked directly. If they wished, participants could create additional nodes, but they were required to create at least one additional node Procedure. Participants were tested individually or in small groups at well separated computers. Problems were presented, one at a time, in a 6 6-inch square in the centre of a standard 16-inch computer display. Participants could begin at any point by left-clicking on a node (or on a location where they wished to establish a new node) with the computer mouse. Then, whilst holding down the mouse button, they drew a path by dragging the mouse cursor to a subsequent node and releasing the button, causing a straight line to be drawn between that node and the previously visited node. By right-clicking on a link (or created node) to select it, and then pressing the Delete key on the keyboard, participants could undo any links or nodes they had drawn. Participants were thus free to connect the nodes in any order, to work alternately from two nodes, or to work on several separated clusters of nodes. Participants signalled when they had completed a problem by clicking on a screen button labelled Proceed to Next Test. At this point, if a participant s completed solution was invalid (e.g., because not all TSP nodes had been connected or because a TSP node had more than two links), a warning message was posted on the screen and the participant was obliged to construct a valid solution before proceeding to the next problem. When they felt ready, participants clicked on the OK button in the message box, and were presented with the next problem. After finishing each type of problem, participants completed a brief questionnaire, before proceeding to the next type of problem.

6 3. Results To permit comparison across the different problem types, participants solution lengths were expressed as a proportion above the optimal (or best known) solution length. This means that performance scores are non-negative, with the optimal solution corresponding to a performance score of zero, and increasingly large performance scores corresponding to increasingly sub-optimal solutions. Performance on the APM Set 2 was measured in terms of the number of correct items out of a total of Comparisons between performance on the TSP, MSTP, GSTP, and APM The Pearson correlations between test 1 and test 2 performance on the TSP, MSTP, and GSTP were all positive (Pearson r = 0.68, 0.78, and 0.49, respectively; N = 48 in all three cases). Accordingly, the two performance scores for each participant were combined to give an average performance score for each test. The means across all participants of these performance scores were 0.08, 0.06, and 0.12 for the TSP, MSTP, and GSTP, respectively. Insert Figure 3 about here Figure 3 shows the relationship between the average performance score and APM scores for each of the three problem types, as well as for each participants average performance across all problem types. Insert Table 1 about here Table 1 presents the Pearson correlations between performance on each pair of problems and between performance on each problem and on the APM. All the coefficients are positive, modest to moderately strong, and highly significant statistically. When the performance measures for all three types of problem were summed, the Pearson correlation with APM was Path complexity as a predictor of performance across the three types of problem In an attempt to focus more closely on a possible common basis for successful performance on the three optimization problems, measures of path complexity were calculated for each participant s solution to each problem. For each link of a participant s solution, numbers (1 to n) were assigned, according to whether the nodes of that link were connected to the nearest (1), second nearest (2), or n th nearest node. The sum of these numbers was divided by the number of links to give a measure of path complexity. The lower the path complexity, the more the participant succeeded in connecting each node to its nearest neighbours (i.e., the more the participant made use of least distances in arriving at his or her solution). Measures of path complexity correlated strongly with average performance for each problem (Pearson r = 0.99, 0.98, and 0.88 for the TSP, MSTP, and GSTP, respectively; N = 48, in each case). Measures of path complexity also showed reliable

7 intercorrelations (Pearson r = 0.64, 0.53, and 0.69, respectively for the TSP:MSTP, TSP:GSTP, and MSTP:GSTP comparisons; N = 48, in each case). In addition, path complexity measures for each type of problem correlated reasonably well with APM scores (Pearson r = -0.50, -0.44, and 0.40 for the TSP, MSTP, and GSTP, respectively; N = 48, in each case) Statistical significance of the correlations Under the Null-Hypothesis Significance Testing (NHST) approach to statistical inference, these correlations between performances on the same test, between average performances on different tests, between test performances and APM scores, and between path complexity measures and performance scores are all highly significant, with p<.01. We are sensitive, however, to criticisms of NHST (e.g., Cohen 1994; Edwards, Lindman, & Savage, 1963; Hunter, 1997). In particular, we acknowledge that NHST violates the likelihood principle, and so, for the reasons explained by Lindley (1972), does not satisfy a basic requirement for rational, consistent and coherent statistical decision making. Accordingly, we also undertook Bayesian analyses of the data (e.g., Lindley, 1972; Sivia, 1996). In particular, we assessed the statistical significance of correlations using Bayes Factors (Kass & Raftery, 1995), which measure how much more (or less) likely one model is than another, on the basis of the evidence provided by data. For correlational analyses, the two models we compared were a null model, which assumed no relationship between the variables, and so assigned data points at random, and an alternative model that assumed the two variables were linearly related. The Bayes Factor for these two models is straightforward to estimate using standard Monte-Carlo techniques (Kass & Raftery, p. 779), involving the sampling of a large number of specific data distributions predicted by the null and alternative models, and calculating their average level of fit to the observed data. For all of the correlations reported above that are significant under NHST, we found that the estimated Bayes Factors showed the alternative model was always more than 1,000 times more likely than the null model. On this basis, it seems safe to conclude that the reported correlation coefficients indicate that a linear relationship between the variables is a better account than one based on random variation, and so the coefficients are statistically significant. 4. Discussion In contrast to the absence of individual differences in the TSP studies of MacGregor and Ormerod (19960 and MacGregor et al. (2000), the results of this study provide further corroboration of the finding by Vickers et al. (2001) and by Vickers, Mayo, et al. (submitted) that performance on visually presented TSP tasks shows reliable individual differences, provided that the problems are sufficiently difficult (i.e., contain a sufficient number of randomly distributed nodes). The present study also replicates the finding by Vickers et al. (2000) and by Vickers, Mayo, et al. (submitted) that TSP performance is reliably associated with performance on a psychometric test of intelligence (Raven s APM). The results of the present study further indicate that there are consistent individual differences in two other, visually presented, optimization tasks (the MSTP and the GSTP), which had been selected because they could be interpreted a priori as,

8 respectively, closer to and further from the process of perceptual organization based on least distances that appears to be an important factor in performance on TSP tasks. Performance on all three tasks was stable across two different instances of the same problem and was reliably correlated across successive instances. In addition, performance on all three tasks was strongly intercorrelated and had equally strong correlations with APM performance. Taken together, these results suggest that performance on the three optimization tasks and on the APM involves an important common factor. The primary question that remains is whether this factor is predominantly perceptual or cognitive. In our view, it seems most likely that the common factor could be identified as the ability to perceive visual organization or structure. As shown by the high correlations between solution length and path complexity measures, as well as the quite strong intercorrelations between path complexity measures, the TSP, MSTP, and GSTP all depend on the ability of participants to utilize visual structure as incorporated in the pattern of least distances between the elements of an array. Nevertheless, there is obviously an additional element associated with the way these least distances are integrated into an overall minimal structure. In the case of the GSTP, in particular, it seemed a priori most likely that this additional element could be interpreted as an abstract cognitive calculation. However, there are some phenomena and theoretical approaches that suggest that perceptual organizing tendencies may also be influenced by other internal structure. For example, there have been several studies of the locations chosen by people when asked to place dots at random within differently shaped outlines (e.g., Lisanby & Lockhead, 1991; Psotka, 1978; Stadler, Richter, Pfaff, & Kruse, 1991). The composite internal structures that emerge in this process show consistencies that have been interpreted in terms of the vectors of underlying force fields (Stadler, Richter, Pfaff, & Kruse, 1991), the extraction of axes of local symmetry (Palmer, 1991), and a so-called grass-fire process for the determination of balanced stick-figures (Blum, 1973; Psotka, 1978). These different possibilities quite closely resemble each other. They are also quite similar to the Generalized Steiner Tree solutions. For example, the optimal interpolated nodes are characterized by 120º angles between their edges. Thus, the edges linked to optimal interpolated nodes have a threefold rotational symmetry, and the visual system is sensitive to the possible presence of most kinds of symmetry (Leyton, 1992; Palmer, 1991; Tyler, 1996). Moreover, because many natural phenomena are regulated by minimization processes (e.g., the hexagonal pattern of a honeycomb, the cracking of dried mud, or the framework of a dragonfly wing), it is often the case that adjacent elements of a surface are linked by similar, threefold rotationally symmetric edges (Hildebrandt & Tromba, 1996). It is possible that this is a form and order of symmetry to which the visual system is particularly sensitive. Therefore, although the GSTP task seems intuitively to involve perceptual processes that are more cognitively directed, it is still possible that its performance is influenced by automatic, locally focussed processes, albeit ones that are concerned with more abstract aspects of organization or the detection of minimal structure. The data on solution lengths and path complexity measures from the present study indicate that the three optimization tasks not only share a common factor, but that this factor also plays an important role in performance on the APM. Once again, it seems most likely that this common factor is a perceptual one. Consistent with this

9 interpretation, DeShon, Chan, and Weissbein (1995) used a verbal overshadowing technique to determine whether performance across all items in the APM depended on the same processes. These authors concluded, in agreement with Hunt (1974), that performance was determined by two general strategies: an analytic one, based on propositional representations, and a visuo-spatial process. The visuo-spatial process appears to be involved in recognizing the nature of various transformations and extrapolating from them. This visuo-spatial process is very similar to the perceptual process that is proposed in Vickers (2001; 2002) generative transformational theory of visual perception (see also Vickers & Preiss, submitted). According to this theory, the visual system uses information about the relative positions of stimulus elements (e.g., in the form of least distances) to home in on those transformations that maximise selfsimilarity in an image (where self-similarity designates the approximate symmetry or correspondence of any part of an image with any other part, including the whole, under a range of possible transformations). The perceptual organization that is seen can be identified with the trajectories of the simplest transformations that maximize a measure of generalized self-similarity. On this view, the processes involved in the perception of organization may be characterized as the detection of various forms of minimal structures, and the same - or closely similar processes appear to be involved in a substantial proportion of the items in the APM. 5. Conclusions The results from the present experiment suggest a number of conclusions. First, they show that, despite earlier indications to the contrary, there are stable and reliable individual differences in performance on visually presented TSP tasks, and that these differences extend to other visually presented optimization tasks that are arguably different in their emphasis on the basic elements of visual structure. Second, the results confirm the findings of two previous studies that showed a relationship between TSP performance and intelligence as assessed by Raven s APM, and they indicate that this relationship is also shared by other related, but distinguishable tasks involving the detection or construction of minimal structures. Third, the present results are consistent with a general, generative transformational theory of visual perception that assumes that the perception of minimal structures may depend on the detection of specific relations among the elements of an array that in turn allow the application of appropriate transformations to maximize the self-similarity in the array. This description also appears to apply quite closely to the solution process involved in a substantial proportion of the APM items. More generally, the present study provides some initial confirmation of the potential of an optimization perspective on intelligence, in which intelligence is seen as the ability to find near-optimal solutions to problems, for which it is impossible (either in principle or in practice) to arrive at a definitive solution through the application of a fixed algorithmic procedure, or for which the computational and/or memory demands exceed the capacity of the brain. The study also raises the question as to how the ability to solve visual optimization problems might be related to performance on other forms of difficult or computationally intractable problem, such as the so-called Secretary Problem, for which there is no natural visual representation (Gilbert & Mosteller, 1966). Meanwhile, in our view, the present results also raise the

10 possibility that either perception is even more intelligent than has been generally assumed or that cognition is more perceptually based.

11 Acknowledgments The research on which this article is based was supported by an Australian Research Council Grant (DP ) to D. Vickers. Correspondence and requests for reprints should be sent to Douglas Vickers, Psychology Department, University of Adelaide, Adelaide, South Australia 5005, or by to

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15 Table TSP MSTP GSTP APM N TSP MSTP GSTP Table 1. Pearson intercorrelations between performance on the three types of problem and APM scores.

16 Figures Figure 1. Two of the three stimulus arrays used in the present experiment. Figure 1a shows the 50-node TSP array and Figure 1b shows the 50-node MSTP array, with the optimal solution in both cases as solid connecting edges. Figure 2. The Generalized Steiner Tree Problem. Figure 2a illustrates the Fermat point (P), as an open circle, and the optimal solution (broken lines) for a simple triangular array. Figure 2b shows the 15-node GSTP array used in the present experiment, together with the interpolated points (open circles) and the optimal tree solution (broken lines) found by a computational algorithm. Figure 3. Scatterplots showing the relationship between APM scores for each participant, and their mean performance on each of the three problems types (TSP, MST, and GSTP), as well as with their average performance across all three problem types.

17 Figure 1. Vickers, Heitman, Lee, & Hughes.

18 Figure 2. Vickers, Heitman, Lee, & Hughes.

19 Performance 0.5 TSP GSTP MTSP ALL APM Figure 3. Vickers, Heitman, Lee, & Hughes.