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Diffusive behavior of a greedy traveling salesman
Adam Lipowski1, Dorota Lipowska
1Faculty of Physics, Adam Mickiewicz University, Poznań, Poland.
Summary
The greedy algorithm
Area of Science:
- Computational mathematics
- Statistical physics
- Operations research
Background:
- The traveling salesman problem (TSP) is a classic combinatorial optimization problem.
- Understanding the diffusive properties of algorithms used to solve TSP is crucial for various applications.
- Anomalous diffusion, such as Lévy flights, is observed in natural systems.
Purpose of the Study:
- To investigate the diffusive properties of the greedy algorithm in the d-dimensional traveling salesman problem.
- To identify potential critical dimensions and scaling behaviors.
- To explore connections between algorithmic strategies and natural foraging behaviors.
Main Methods:
- Monte Carlo simulations were employed to model the greedy algorithm's behavior.
- The average squared distance from the origin (r^2) was analyzed as a function of the number of steps (t).
- Distributions of path lengths were examined across different dimensions.
Main Results:
- For dimensions d=3 and d=4, the average squared distance scales linearly with the number of steps (r^2 ∝ t).
- In the d=2 case, logarithmic corrections modify this linear scaling, suggesting d=2 as a critical dimension.
- Distinct differences in length distributions were observed between d=2 and d>2.
Conclusions:
- The greedy algorithm's diffusive behavior in TSP is dimension-dependent.
- The observed scaling and distributions may offer insights into anomalous diffusion in natural systems like foraging.
- Dimension-dependent search space properties could explain broad and Lévy-like distributions in such systems.
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