The Role of Problem Representation in Producing Near-Optimal TSP Tours

Abstract

Gestalt psychologists pointed out about 100 years ago that a key to solving difficult insight problems is to change the mental representation of the problem, as is the case, for example, with solving the six matches problem in 2D vs. 3D space. In this study we ask a different question, namely what representation is used when subjects solve search, rather than insight problems. Some search problems, such as the traveling salesman problem (TSP), are defined in the Euclidean plane on the computer monitor or on a piece of paper, and it seems natural to assume that subjects who solve a Euclidean TSP do so using a Euclidean representation. It is natural to make this assumption because the TSP task is defined in that space. We provide evidence that, on the contrary, subjects may produce TSP tours in the complex-log representation of the TSP city map. The complex-log map is a reasonable assumption here, because there is evidence suggesting that the retinal image is represented in the primary visual cortex as a complex-log transformation of the retina. It follows that the subject's brain may be "solving" the TSP using complex-log maps. We conclude by pointing out that solving a Euclidean problem in a complex-log representation may be acceptable, even desirable, if the subject is looking for near-optimal, rather than optimal solutions.

When is Psychology Research Useful in Artificial Intelligence? A Case for Reducing Computational Complexity in Problem Solving

“…moment associated with the use of clustering). Second, the representation of the problem that humans solve is different from the problem that is posed to them (Fleischer et al., 2018). In the TSP, participants are asked to find the shortest tour, which requires calculating the lengths of all possible tours.…”

“…The most common way to represent problem-solving scenarios is to specify a graph in which one or more nodes are starting states (e.g., all the information about the environment 2 available before beginning to solve the problem), one or more nodes are goal states (e.g., all the information about the environment available after solving the problem), and edges represent transitions between pairs of states (operators) (Fleischer, Hélie, & Pizlo, 2018;Newell & Simon, 1972). Each edge has a cost.…”