Simplicity and Complexity in Combinatorial Optimization

Area of Science:

• Theoretical Computer Science
• Mathematical Physics
• Optimization Theory

Background:

• Combinatorial optimization problems are prevalent in physics and computer science.
• Understanding the theoretical underpinnings of optimization is crucial for advancing these fields.

Purpose of the Study:

• To investigate the relationship between Kolmogorov complexity and the properties of optima in optimization problems.
• To explore the potential of algorithmic probability for optimization.
• To analyze the likelihood of coincidences in optimization problem extrema.

Main Methods:

• Theoretical analysis connecting Kolmogorov complexity with optimization optima.
• Examination of optimization via sampling candidate solutions based on algorithmic probability.
• Statistical analysis of coincidences in extrema compared to a random null model.

Main Results:

• A theoretical connection is established between optima and complexity, indicating extrema are often low-complexity.
• Optimization using algorithmic probability sampling is proposed as a potentially effective method.
• Coincidences in optimization problem extrema are shown to be more probable than under a random model.

Conclusions:

• Kolmogorov complexity provides insights into the nature of optimization optima.
• Algorithmic probability offers a novel approach to optimization.
• The non-random nature of extrema in optimization problems has significant theoretical implications.