Computation of the travelling salesman problem by a shrinking blob

Abstract

The travelling salesman problem (TSP) is a well known and challenging combinatorial optimisation problem. Its computational intractability has attracted a number of heuristic approaches to generate satisfactory, if not optimal, candidate solutions. Some methods take their inspiration from natural systems, extracting the salient features of such systems for use in classical computer algorithms. In this paper we demonstrate a simple unconventional computation method to approximate the Euclidean TSP using a virtual material approach. The morphological adaptation behaviour of the material emerges from the low-level interactions of a population of particles moving within a diffusive lattice. A ‘blob’ of this material is placed over a set of data points projected into the lattice, representing TSP city locations, and the blob is reduced in size over time. As the blob shrinks it morphologically adapts to the configuration of the cities. The shrinkage process automatically stops when the blob no longer completely covers all cities. By manually tracing the perimeter of the blob a path between cities is elicited corresponding to a TSP tour. Over 10 runs on 20 randomly generated datasets consisting of 20 cities this simple and unguided method found tours with a mean average tour length of 6.41 % longer than the minimum tours computed by a TSP solver (mean best performance was 4.27 % longer and mean worst performance was 9.22 % longer). We examine the insertion mechanism by which the blob constructs a tour, note some properties and limitations of its performance, and discuss the relationship between the blob TSP and proximity graphs which group points on the plane. The method is notable for its simplicity, novelty and the spatially represented mechanical mode of its operation. We discuss similarities between this method and previously suggested models of human performance on the TSP and suggest possibilities for further improvement.

Appendix: Shrinking blob particle model description

The multi-agent particle approach to generate the behaviour of the virtual material blob uses a population of coupled mobile particles with very simple behaviours, residing within a 2D diffusive lattice. Lattice size was 200 × 200 pixels. The lattice stores particle positions and the concentration of a local diffusive factor referred to generically as chemoattractant. Collective particle positions represent the global pattern of the blob. The particles act independently and iteration of the particle population is performed randomly to avoid any artifacts from sequential ordering.

1.1 Generation of emergent blob cohesion and shape adaptation

The behaviour of the particles occurs in two distinct stages, the sensory stage and the motor stage. In the sensory stage, the particles sample their local environment using three forward biased sensors whose angle from the forwards position (the sensor angle parameter, SA), and distance (sensor offset, SO) may be parametrically adjusted (Fig. 13a). The offset sensors generate local coupling of sensory inputs and movement to generate the cohesion of the blob. The SO distance is measured in pixels and a minimum distance of 3 pixels is required for strong local coupling to occur. During the sensory stage each particle changes its orientation to rotate (via the parameter rotation angle, RA) towards the strongest local source of chemoattractant (Fig. 13b). After the sensory stage, each particle executes the motor stage and attempts to move forwards in its current orientation (an angle from 0–360°) by a single pixel forwards. Each lattice site may only store a single particle and particles deposit chemoattractant into the lattice only in the event of a successful forwards movement. If the next chosen site is already occupied by another particle move is abandoned and the particle selects a new randomly chosen direction.

Fig. 13

Architecture of a single component of the shrinking blob and its sensory algorithm. a Morphology showing agent position ‘C’ and offset sensor positions (FL, F, FR), b algorithm for particle sensory stage

1.2 TSP problem representation

Twenty datasets were generated, each consisting of 20 randomly chosen data points within a circular arena. A condition was added that a minimum distance of 25 pixels must exist between data points. This gives a more uniform distribution of node points, preventing clustering of node points often found in real-world TSP instances, for example those based on real city locations. This was done partly to aid visual tracking of the tour path but also because a less clustered node distribution is known to provide more challenging problem instances (e.g. in human performance on TSP (Hirtle and Gärling 1992)) since there is less likelihood of providing pre-existing cues to intuitive solutions (for example nearest neighbour grouping). Each dataset was saved to a text file. TSP city data points for each dataset were loaded from the text files and were represented by projection of chemoattractant to the diffusion lattice at locations corresponding to a 3 × 3 window centred about their x, y position. The projection concentration was 1.275 units per scheduler step. If a node was covered by a portion of the blob (i.e. if the number of agents in a 3 × 3 window surrounding the node was >0) the projection was reduced to 0.01275 units. Suppression of projection from covered sites was necessary to ensure a uniform concentration within the blob at internal data points. Uncovering of the data points by the shrinking blob acted to increase concentration at exposed nodes, causing the blob to be anchored by the nodes. Diffusion in the lattice was implemented at each scheduler step and at every site in the lattice via a simple mean filter of kernel size 3 × 3. Damping of the diffusion distance, which limits the distance of chemoattractant gradient diffusion, was achieved by multiplying the mean kernel value by 0.95 per scheduler step.

The blob was initialised by creating a population of particles and inoculating the population within the bounds of a Convex Hull formed by the data points, generate at the start of the experiment by a conventional algorithm. Why choose the Convex Hull as the initial blob pattern? It is possible to select any pattern for the blob shape, for example a circular blob. However, selecting any arbitrary pattern to cover the points might leave a large portion of the blob overlapping vacant space. This could result in shrinkage in one part of the blob taking place over vacant space with shrinkage in other parts over city points and would bias the output result. The Convex Hull was thus chosen as a simple representation of the perimeter of the data points. The exact population size differed depending on the area of the Convex Hull but the initial blob was typically composed of between 10,000 and 15,000 particles. Particles were given random initial positions within these confines and random initial orientations. Any particles migrating out of the bounds of the Convex Hull region were deleted. Particle sensor offset (SO) was 7 pixels. Angle of rotation (RA) and sensor angle (SA) were both set to 60° in all experiments. Agent forward displacement was 1 pixel per scheduler step and particles moving forwards successfully deposited five units of chemoattractant into the diffusion lattice. Both data projection stimuli and agent particle trails were represented by the same chemoattractant ensuring that the particles were attracted to both data stimuli and other agents’ trails. The collective behaviour of the particle population was cohesion and morphological adaptation to the configuration of stimuli.

1.3 Shrinkage mechanism

Adaptation of the blob size is implemented via tests at regular intervals as follows. If there are 1–10 particles in a 9 × 9 neighbourhood of a particle, and the particle has moved forwards successfully, the particle attempts to divide into two if there is a space available at a randomly selected empty location in the immediate 3 × 3 neighbourhood surrounding the particle. If there are 0–80 particles in a 9 × 9 neighbourhood of a particle the particle survives, otherwise it is deleted. Deletion of a particle leaves a vacant space at this location which is filled by nearby particles, causing the blob to shrink slightly. As the process continues the blob shrinks and adapts to the stimuli provided by the configuration of city data points. The frequency at which the growth and shrinkage of the population is executed determines a turnover rate for the particles. The frequency of testing for particle division was every five scheduler steps and the frequency for testing for particle removal was every ten scheduler steps. Since the shrinking blob method is only concerned with the reduction in size of the population it might be asked as to why there are tests for particle division at all. The particle division mechanism is present to ensure that the adaptation of the blob sheet is uniform across the sheet to prevent ‘tears’ or holes forming within the blob sheet, particularly at the start of an experiment when flux within the blob is initially established.

1.4 Halting mechanism

The shrinkage of the blob was halted when all data points were partially uncovered by using the following calculation. At each data point the number of particles surrounding the point in a 5 × 5 window was sampled. If the number of particles in this window was <15 the node was classified as partially uncovered. When all nodes are uncovered the model is halted and a greyscale representation of the diffusive lattice is saved to disk to read the result of the blob tour. The calculated tour path was saved in a text file and tour distance was calculated. The exact minimum tour path for each dataset was calculated for comparison by loading the data point configuration files into the Concorde TSP solver (Applegate et al. 2006).