Running head: CONVEX HULL AND TOUR CROSSINGS IN TSP Convex Hull and Tour Crossings in the Euclidean Traveling Salesperson Problem: Implications for Human Performance Studies

Recently there has been growing interest among psychologists in human performance on the Euclidean Traveling Salesperson problem (E-TSP). A debate has been initiated on what strategy people use in solving visually presented E-TSP instances. The most prominent hypothesis is the convex-hull hypothesis, originally proposed by MacGregor and Ormerod (1996). We argue that, in the literature so far, there is no evidence for this hypothesis. Alternatively we propose and motivate the hypothesis that people aim at avoiding crossings. I. van Rooij Convex hull and tour crossings in TSP 3 Convex Hull and Tour Crossings in the Euclidean Traveling Salesperson Problem: Implications for Human Performance Studies The Traveling Salesperson problem (TSP; traditionally referred to as the Traveling Salesman problem) can be informally described as follows. For a given set of points and costs between each pair of points, find a cheapest tour that visits each point exactly once. In the Euclidean version of the Traveling Salesperson problem (E-TSP) the points are positioned in the Euclidean plane and the costs are the distances implied by the Euclidean geometry. The study presented by MacGregor and Ormerod (1996) is one of the first psychological studies that explicitly and primarily focuses on human performance on E-TSP (but see also Polivanova, 1974). Their study has motivated further psychological research on the topic, and has initiated a debate about what strategy people use in solving visually presented E-TSP instances (Graham, Joshi, & Pizlo, 2000; MacGregor, Ormerod, & Chronicle, 1999, 2000; Ormerod & Chronicle, 1999; Vickers, Butavicius, Lee, & Medvedev, 2001; see also Lee & Vickers, 2000, for a commentary of MacGregor & Ormerod, 1996, and see MacGregor & Ormerod, 2000, for a reply). MacGregor and Ormerod (1996) proposed the convex-hull hypothesis, which states that people use the convex hull as part of their strategy to construct E-TSP tours. The convex hull of a set of points in the plane is the smallest convex polygon that encloses all the points in the set (the points on the convex hull are also called boundary points and the remaining points are then called interior points; see also Figure 1). The convex-hull hypothesis is based on the assumption that (1) people can identify the convex hull of a point set in the plane via an automatic perceptual process and (2) people are sensitive to the principle that optimal tours connect points on the I. van Rooij Convex hull and tour crossings in TSP 4 convex hull in order of adjacency (see also MacGregor et al., 1999, 2000; Ormerod & Chronicle, 1999). The purpose of this commentary is to point out the absence of evidence for the convexhull hypothesis in the literature on human performance on E-TSP. We further suggest an alternative hypothesis called the crossing-avoidance hypothesis. A crossing-free tour that respects the convex hull is no surprise MacGregor and Ormerod (1996) find that tours proposed by human participants tend to connect convex-hull points in order. They claim that this finding provides evidence for the convex-hull hypothesis (see also Graham et al., 2000; MacGregor et al., 1999, 2000; Ormerod & Chronicle, 1999). We show that, because tours produced by participants seldom contain crossings, this conclusion is not warranted. ---------------------------------------------------------insert figure 1 about here ---------------------------------------------------------We start by stating two observations. Observation 1a. A tour that does not follow the convex hull contains at least one crossing. For an illustration of a proof refer to Figure 1. Note that Observation 1a is equivalent to Observation 1b. Observation 1b. A tour that does not contain any crossings follows the convex hull. Observation 2. There exist tours with at least one crossing that follow the convex hull. Consistent with Observation 1 and 2, MacGregor and Ormerod (1996, pp. 531-532) write: “Failure to connect boundary points in order of adjacency automatically creates a solution with crossed arcs. However, crossed arcs can also occur when boundary points are connected in I. van Rooij Convex hull and tour crossings in TSP 5 order.” Inconsistent with Observation 1, they go on to report that “Almost invariably, subjects connected boundary points in order of adjacency and, equally invariably produced no crossed arcs” (1996, p. 536). This last sentence seems to state two independent findings. In a similar vein, MacGregor et al. (2000, p. 1184) enumerate the main findings of MacGregor and Ormerod (1996) with one point being “people produce paths that connect boundary points in sequential order of adjacency (449 of 455 solutions adhered to this principle),” and another, again seemingly independent point being “few solutions have lines that cross (11 of the 455 solutions).” Note, however, that the fact that 444 of the proposed tours contain no crossing logically implies that these 444 tours visit the boundary points in order of adjacency (Observation 1). Therefore the finding that the majority of the subjects connect boundary points in order of adjacency is no surprise, but a logical consequence of the fact that the majority of tours have no crossings. In other words, these two findings are not independent empirical findings. Do tours with crossings follow the convex-hull? Observation 1 implies that we have to consider tours with crossings to provide evidence for the hypothesis that people aim at following the convex hull (see Table 1). Table 2 gives an overview of the number (proportion) of tours with crossings that follow the convex hull, for Experiment 1 and 2 of MacGregor and Ormerod (1996), Experiment 1, 2 and 3 of MacGregor et al. (1999), the optimization group (Group O) in Experiment 1 of Vickers et al. (2001), and the adult group in the experiment by Schactman (2002). ---------------------------------------------------------insert tables 1 and 2 about here ---------------------------------------------------------I. van Rooij Convex hull and tour crossings in TSP 6 For ease of presentation the data in Table 2 are collapsed over instances and participants within each experiment. We note that the occurrence of crossings does not seem to be strictly instance specific, nor does the failure to avoid crossings seem to be strictly participant specific. Namely, for 3 of the 6 instances in Experiment 1, for 4 of the 7 instances in Experiment 2 of MacGregor and Ormerod (1996), for 5 of the 6 instances of Vickers et al. (2001), and for 8 of the 15 instances of Schactman (2002), at least one tour with crossings was observed. The tours with crossings in Experiment 1 and 2 of MacGregor and Ormerod (1996) were produced by 5 and 1 participants respectively, in Vickers et al. (2001) were produced by 7 participants, and in Schactman (2002) were produced by 4 participants. Table 2 shows that overall about 38% of the tours with crossings visit the convex-hull points in order of adjacency. From Table 2 we can conclude that there is no evidence of a tendency to follow the convex hull. Is there evidence for the convex-hull hypothesis in the literature? In the literature seven findings are presented as evidence for the convex-hull hypothesis (MacGregor & Ormerod, 1996; MacGregor et al., 1999, 2000): (1) people tend to follow the convex hull, (2) response uncertainty is a function of the number of interior points, (3) people tend to produce tours without crossings, (4) people tend to produce tours with relatively few indentations (an indentation in a tour occurs if at least one interior point is visited between two boundary points), (5) performance is better when interior points are located relatively close to the convex hull than when they are located far away from the convex hull, (6) tours produced by a convex-hull heuristic (cf. MacGregor et al., 2000) are close in length to tours produced by humans, and (7) this heuristic’s performance is qualitatively similar to human performance. We I. van Rooij Convex hull and tour crossings in TSP 7 have argued above that finding (1) does not provide evidence for the convex-hull hypothesis. In the following we argue that this also holds for findings (2) through (7). As MacGregor and Ormerod argued (1996, p. 528), if the tour is constrained to follow the convex hull then decreasing the number of interior points (while keeping the total number of points fixed) reduces the degrees of freedom for connecting the points. We have shown that the absence of crossings implies a tour that follows the convex hull. Thus, finding (2) is also a consequence of finding (3). Finding (3) does not provide evidence for the convex-hull hypothesis, simply because following the convex-hull points in order of adjacency does not prevent crossings (Observation 2). As pointed out by MacGregor and Ormerod (1996; see also Lee & Vickers, 2000) and MacGregor et al. (2000), given the highly constrained nature of the stimuli used, the findings (4) and (5) may be artefacts. Because the optimal tours for the instances used by MacGregor and Ormerod (1996) all had very few indentations (see e.g. Figure 1 in Ormerod & Chronicle, 1999) finding (4) follows directly from the close to optimal performance by participants on these instances. We further note that finding (4) is unlikely to be replicated with random instances, since connecting boundary points to each other is in general clearly a non-optimal strategy. To argue finding (6) does not provide evidence for the convexhull hypothesis we note that human performance in the considered experiments is close to optimal. Any heuristic that models human performance will have to produce close to optimal tours (and thus close to human performance). Thus comparing only lengths of tours produced by humans and heuristics is not informative about the strategy of humans. We studied the convexhull heuristic (cheapest insertion criterion) described by MacGregor et al. (2000, pp. 1184-1186) and disagree that it qualitatively models human behaviour (finding (7)). MacGregor et al. distinguish between sketched and connected arcs. The heuristic allows the construction of a I. van Rooij Convex hull and tour crossings in TSP 8 closed path consisting of connected arcs only, even before all interior points are included in the tour (see Figure 2 for an illustration). As a consequence, it can happen that the heuristic visits certain points (and arcs) several times before the tour is completed. Also, previously connected arcs may need to be disconnected and replaced by a subpath in order to complete the tour. This behaviour of the heuristic is inconsistent with the motivation of MacGregor et al. to model the sequential character of tour construction by humans. ---------------------------------------------------------insert figure 2 about here ---------------------------------------------------------The crossing-avoidance hypothesis We have shown that the low proportion of tours with crossings begs for an explanation, not the high proportion of tours that respect the convex hull (see Observation 1). As said, the convex-hull hypothesis does not explain the low proportion of tours with crossings (see Observation 2). Therefore, we propose the crossing-avoidance hypothesis. This hypothesis states that humans aim to avoid crossed lines in the plane when trying to solve E-TSP, because they are sensitive to the fact that tours with crossed lines are non-optimal. Here we argue, the observation that crossings are non-optimal is more elementary than the observation that optimal tours follow the convex hull. Consequently, it seems more plausible to assume that people are, at least, sensitive to the fact that optimal tours have no crossings. We note the following observation. Observation 3. A tour that contains at least one crossing is non-optimal (Flood, 1956). I. van Rooij Convex hull and tour crossings in TSP 9 Notice that Observation 3 follows directly from the triangle inequality that holds in the Euclidean plane (see Figure 3). Simply put, a crossing is non-optimal because one can always replace the two crossed arcs by two shorter ones. From Observation 1 and 3 we can conclude: Corollary 1. An optimal tour follows the convex hull (Golden, Bodin, Doyle, & Stewart, 1980). ---------------------------------------------------------insert figure 3 about here ---------------------------------------------------------We have shown the property that an optimal tour does not contain any crossings is indeed more elementary than the property that an optimal tour follows the convex hull because the latter derives from the former. MacGregor et al. (2000, p. 1184) state the issue backwards when they write “the optimal path connects adjacent points on the boundary of the convex hull in sequence (...) A corollary is that solutions that do not adhere to this principle will result in crossed arcs, which is clearly non-optimal.” The fact that tours that do not follow the convex hull have crossings is in no sense a corollary of the fact that optimal tours follow the convex hull. Instead the fact that optimal tours follow the convex hull is implied by Observation 1 and 3. Note that MacGregor et al. state explicitly that crossings are clearly non-optimal (see also Flood, 1956, p. 64). Also Ormerod and Chronicle, who performed experiments in which participants had to judge whether a pre-drawn tour was optimal or not, considered crossings as indicators of nonoptimality (Ormerod & Chronicle, 1999, p. 1236). The non-optimality of tours with crossings seems intuitive, because it follows directly from a very basic and visually transparent property of the Euclidean plane; i.e., the shortest path between two points is a straight line. Therefore it is plausible to assume that people try to avoid crossings when searching for a shortest path or tour. I. van Rooij Convex hull and tour crossings in TSP 10 The crossing-avoidance hypothesis has another advantage: It is more generally applicable than the strategy of following the convex-hull. As pointed out by Graham et al. (2000, p. 1197), a convex-hull strategy is only potentially useful in E-TSP, but not in “other spatial problems, such as, finding a shortest path between start and a goal.” Clearly, crossing-avoidance is also a good strategy in the latter situation, as well as for problems like Minimum Euclidean Spanning Tree and Minimum Euclidean Steiner Tree (see e.g. Garey & Johnson, 1979, for problem definitions). We briefly comment on a possible problem for interpretation of the low proportion of tours with crossings that occurred to us when inspecting the raw data from previous experiments. We noticed that some participants produce tours with crossings that were not crossings in a physical sense. That is, a straight line between the two points in question would have produced a crossing, but the participant had chosen to make a detour (i.e., a curved line) so that no physical crossing occurred. The last column in Table 2 lists the frequency of occurrences of such “nonphysical crossings” for each study. The observation of such non-physical crossings suggests that some participants may misconstrue their task as being the task of finding a tour without lines that cross in the plane. Such task misconstruals are common dangers in psychological experiments, and the formulation of E-TSP certainly allows for misinterpretation on the participant’s side. For example, a participant may (unconsciously) infer that s/he is not supposed to make crossings from the fact that the experimenter did not explicitly say that crossings were allowed. Also, it is possible that for some people the word “tour” may have the connotation of being a closed path without any crossings. This is not to say that if people would understand their task properly that they would not still tend to avoid crossings (we believe they would). It just draws attention to the importance of making sure that the task is properly understood, to prevent unwarranted attribution of insightfulness on the part of participants. I. van Rooij Convex hull and tour crossings in TSP 11 Conclusion The convex-hull hypothesis proposes that people aim at following the convex hull when attempting to solve E-TSP instances. One of the main findings presumed to provide support for this hypothesis is the finding that most tours produced by people follow the convex-hull (MacGregor & Ormerod, 1996; MacGregor et al., 1999, 2000). We have shown that this finding does not provide any evidence for the idea that people aim at following the convex hull given the fact that most tours produced by people are crossing-free. Namely, a tour that is crossing-free by definition follows the convex hull. The only way to show that tours produced by people follow the convex hull because of a reason other than being crossing-free is to inspect specifically tours with crossings. We have shown in Table 2 that there is no evidence that such tours tend to follow the convex-hull. We also discussed the problematic nature of the other findings that MacGregor and Ormerod (1996) and MacGregor et al. (1999, 2000) claimed are (indirect) support for the convex-hull hypothesis. Contrary to what these authors suggest, we conclude that at the present time there is no evidence for the convex-hull hypothesis. Finally, we argued that the low proportion of observed tours with crossings invites our crossing-avoidance hypothesis. This hypothesis proposes that people aim at avoiding crossings when attempting to solve E-TSP instances. Unlike the convex-hull hypothesis, the crossingavoidance hypothesis explains the observation that most of the tours produced by people are crossing-free. We further argued that this hypothesis is, a priori, at least as plausible as the convex-hull hypothesis. Of course, the convex-hull hypothesis and crossing-avoidance hypothesis are not mutually exclusive. Some preliminary evidence for both the convex-hull hypothesis and the crossing-avoidance hypothesis may be found in verbal reports by participants in the studies by I. van Rooij Convex hull and tour crossings in TSP 12 Polivanova (1974) and Vickers et al. (2001). To further test these hypotheses, online monitoring of human strategies as well as verbal protocols may prove more useful. I. van RooijConvex hull and tour crossings in TSP 13 ReferencesFlood, M. M. (1956). The traveling-salesman problem. Operations Research, 4, 61-75.Garey, M. R. & Johnson, D. S. (1979). Computers and intractability: A guide to thetheory of NP-completeness. New York: Freeman.Golden, B. L., Bodin, L. D., Doyle, T., Stewart, W. (1980). Approximate travelingsalesman algorithms. Operations Research, 28, 694-711.Graham, S. M., Joshi, A., & Pizlo, Z. (2000). The traveling salesman problem: Ahierarchical model. Memory & Cognition, 28(7), 1191-1204.Lee, M. D. & Vickers, D. (2000). The importance of the convex hull for humanperformance on the traveling salesman problem: A comment on MacGregor and Ormerod(1996). Perception & Psychophysics, 62(1), 226-228.MacGregor, J. N. & Ormerod, T. C. (1996). Human performance on the travelingsalesman problem. Perception & Psychophysics, 58(4), 527-539.MacGregor, J. N. & Ormerod, T. C. (2000). Evaluating the importance of the convex hullin solving the Euclidean version of the traveling salesperson problem: Reply to Lee and Vickers(2000). Perception & Psychophysics, 62(7), 1501-1503.MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (1999). Spatial and contextualfactors in human performance on the traveling salesperson problem. Perception, 28, 1417-1427.MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (2000). A model of human performance on the traveling salesperson problem. Memory & Cognition, 28(7), 1183-1190.Ormerod, T. C. & Chronicle, E. P. (1999). Global perceptual processing in problem solving: The case of the traveling salesperson. Perception & Psychophysics, 61(6), 1227-1238. I. van RooijConvex hull and tour crossings in TSP 14 Polivanova, N. I. (1974). On some functional and structural features of the visual-intuitive components of a problem-solving process. Voprosy Psikhologii [Questions inPsychology], 4, 41-51.Schactman, A. (2002). Children’s performance on the traveling salesperson problem.Unpublished Honours thesis, University of Victoria, Victoria, British Columbia, Canada.Vickers, D., Butavicius, M., Lee, M. D., & Medvedev, A. (2001). Human performance onvisually presented traveling salesman problems. Psychological Research, 65, 34-45. I. van RooijConvex hull and tour crossings in TSP 15 Author NoteWe thank Jim MacGregor, Tom Ormerod, Ed Chronicle, Douglas Vickers, Marcus Butavicius,Michael Lee and Andrei Medvedev for sharing their data. Special thanks to Jim MacGregor,Evan Heit, Tom Ormerod, Michael Lee and an anonymous reviewer for their helpful commentsand suggestions. This research was supported by the National Science and Engineering ResearchCouncil (NSERC) of Canada and a research grant from the University of Victoria, awarded toUlrike Stege. While working on this article Iris van Rooij was additionally supported by NSERCGrant OGP0121280-97 awarded to Helena Kadlec. Correspondence should be addressed to Irisvan Rooij, Department of Psychology, P.O. Box 3050, University of Victoria, Victoria, B.C.,V8W 3P5, Canada (e-mail: [email protected]). I. van RooijConvex hull and tour crossings in TSP 16 Footnotes1 The word “instance” is used to refer to a particular instantiation of the E-TSP problem (i.e., aparticular set of points for which one is to find the shortest tour), whereas the word “problem” isused to refer to the generic problem, E-TSP (i.e., given a set of points find a shortest tour).2 Also revisiting a point, traversing an edge more than once, or a non-physical crossing wasscored as a crossing.3 In Schactman (2002) 18 adults were presented with 15 random point sets, with 5, 10 or 15points. Point sets were presented on paper and participants were instructed to draw the shortestpossible tour through the set of points. Performance of the adult group was comparable to thatobserved in previous studies. That is, mean deviation from optimal (defined as, [observed tourlength – optimal tour length] / optimal tour length) was overall low, and increased with problemsize (M = 0.016, M = 0.020, and M = 0.027, for the 5-, 10-, and 15-point instances respectively).4 Interestingly, the lowest proportion of tours with crossings (only 1 out of 1260 tours) was foundby Graham et al. (2000) in an experiment in which it was impossible for participants to drawcurved lines (i.e., Graham et al. used a computer interface which only allowed people to connectpoints by straight lines). I. van RooijConvex hull and tour crossings in TSP 17 Table 1The three conjunctions of findings for tours crossings (Yes/No) and tours following the convex hull (Yes/No) that are possible. For each conjunction it is indicated whether it constitutesevidence for (+), evidence against (–), or evidence neither for nor against (n) the convex-hull hypothesis and/or the crossing-avoidance hypothesis. Tour hascrossingsTour followsconvex hullConvex-hullhypothesisCrossing-avoidancehypothesis YesYes+–YesNo––NoYesn+ I. van RooijConvex hull and tour crossings in TSP 18 Table 2Descriptives for tours with crossings from previous research. Total numberof instancesNumber (proportion)of tours with crossingsNumber (proportion)of tours with crossingsthat follow convex hullNumber of tourswith non-physicalcrossings MacGregor & Ormerod (1996) Experiment 1 3155 (0.02)0 (0.00)0 Experiment 2 1406 (0.04)5 (0.83)0 MacGregor et. al. (1999) Experiment 1 10320 (0.19)6 (0.30)11 Experiment 2 342 (0.06)1 (0.50)0 Experiment 3 85665 (0.08)23 (0.35)4 Vickers et al. (2001) Experiment 1 (group O) 1089 (0.08)3 (0.33)2