Computation of the Travelling Salesman Problem by a Shrinking Blob (1303.4969v2)

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. 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 6 runs on 20 randomly generated datasets of 20 cities this simple and unguided method found tours with a mean best tour length of 1.04, mean average tour length of 1.07 and mean worst tour length of 1.09 when expressed as a fraction of the minimal tour computed by an exact TSP solver. 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 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.

• The paper introduces a unique shrinking blob method that mimics plasmodium behavior to tackle the Travelling Salesman Problem.
• It employs a multi-agent system on a diffusive lattice where a blob adapts its shape and gradually uncovers TSP nodes.
• Results on randomized 20-city datasets yield mean tour lengths close to optimal, underscoring the method's potential in unconventional computing.

Computation of Travelling Salesman Problem by a Shrinking Blob

The paper "Computation of Travelling Salesman Problem by a Shrinking Blob," authored by Jeff Jones and Andrew Adamatzky, investigates an unconventional approach to solving the Travelling Salesman Problem (TSP), which is a quintessential combinatorial optimization challenge. Their approach revolves around the use of a virtual material that mimics the behavior of the plasmodium Physarum polycephalum, although the authors clarify that the material doesn't naturally conform to TSP network structures. The methodology presented here reflects a departure from traditional computational techniques, leveraging properties of morphological adaptation and minimization to derive potential solutions for the Euclidean TSP.

Methodology

The core of this research is centered on a "shrinking blob" method that employs a virtual material composed of a vast population of simple mobile multi-agent particles within a 2D diffusive lattice. Each particle interacts with a chemoattractant, creating emergent properties of cohesion and adaptive shape minimization. Initially, a blob of this material is placed over TSP city locations projected into the lattice. As time progresses, the blob reduces in size, morphologically adapting to the spatial configuration of the cities. The shrinkage stops when all cities are partially uncovered, allowing the perimeter of the blob to dictate a tour path, revealing a TSP solution.

Results

Upon executing this method over six runs on 20 randomly generated datasets, each consisting of 20 cities, the authors report the method's performance in terms of tour length relative to an exact TSP solver. The blob approach resulted in a mean best tour length of 1.04, mean average tour length of 1.07, and mean worst tour length of 1.09, calculated as fractions of the minimal tour length. These figures affirm the method's simplicity over its computational performance, which is pivotal for approaching larger datasets or more complex configurations.

Discussion of Concave Insertion Process

One noteworthy aspect detailed in the paper is the tour's construction during the blob's shrinkage phase. Initially reflecting a Convex Hull shape, the blob dynamically deforms into a Concave Hull as particles are removed. This gradual adaptation involves inserting nodes into the tour as the blob conformation evolves, aligning with the unique regional concavities of the dataset. Variations in node distance impact the predominance of concavities and subsequently determine the accuracy and efficiency of the calculated tour.

Implications and Future Directions

The results of this research enhance our understanding of unconventional computing methods, showcasing alternative paradigms that contrast sharply with historically heuristic or algorithmic approaches to the TSP. While not superior in computational performance, the insights into emergent material properties provide fertile ground for exploring physical systems' potential in solving mathematical problems. There is scope for further refinement and examination of agent behaviors in different spatial contexts, possibly integrating this virtual material computation within hybrid frameworks that can capitalize on both its simplicity and dynamic adaptation.

In summary, the authors present a compelling investigation into unconventional computational techniques for TSP, emphasizing simplicity and potential insights into human cognitive approaches to similar problems. This paper contributes to the growing body of work in unconventional computing, highlighting the delicate balance between innovative theory and practical application.