Chemotaxis can provide biological organisms with good solutions to the travelling salesman problem

The ability to find good solutions to the traveling salesman problem can benefit some biological organisms. Bacterial infection would, for instance, be eradicated most promptly if cells of the immune system minimized the total distance they traveled when moving between bacteria. Similarly, foragers would maximize their net energy gain if the distance that they traveled between multiple dispersed prey items was minimized. The traveling salesman problem is one of the most intensively studied problems in combinatorial optimization. There are no efficient algorithms for even solving the problem approximately (within a guaranteed constant factor from the optimum) because the problem is nondeterministic polynomial time complete. The best approximate algorithms can typically find solutions within 1%-2% of the optimal, but these are computationally intensive and can not be implemented by biological organisms. Biological organisms could, in principle, implement the less efficient greedy nearest-neighbor algorithm, i.e., always move to the nearest surviving target. Implementation of this strategy does, however, require quite sophisticated cognitive abilities and prior knowledge of the target locations. Here, with the aid of numerical simulations, it is shown that biological organisms can simply use chemotaxis to solve, or at worst provide good solutions (comparable to those found by the greedy algorithm) to, the traveling salesman problem when the targets are sources of a chemoattractant and are modest in number (n < 10). This applies to neutrophils and macrophages in microbial defense and to some predators.