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Trajectories in phase diagrams, growth processes, and computational complexity: how search algorithms solve the
1CNRS-Laboratoire de Physique Théorique de l'ENS, 24 rue Lhomond, 75005 Paris, France.
Physical Review Letters
|April 6, 2001
Summary
Tracking problem parameters during algorithm runs reveals distinct phases, enabling prediction of computational difficulty. This approach helps identify easy or hard problem instances for optimization and decision-making.
Area of Science:
- Computer Science
- Computational Complexity
- Artificial Intelligence
Background:
- Decision and optimization problems are often classified as computationally easy or hard.
- Understanding problem difficulty is crucial for efficient algorithm design and application.
Purpose of the Study:
- To investigate methods for predicting the computational difficulty of problem instances during algorithm execution.
- To analyze the behavior of search algorithms on representative hard problems.
Main Methods:
- Tracking characteristic problem parameters during algorithm execution to define trajectories in parameter space.
- Analyzing these trajectories for phase transitions in the context of 3-satisfiability problems.
- Correlating trajectory behavior with instance difficulty and resolution time.
Main Results:
- Identified well-defined phases within the parameter space trajectories.
- Demonstrated that these phases correspond to domains of easy or hard problem instances.
- Showed successful prediction of resolution times based on trajectory analysis.
Conclusions:
- Algorithm run trajectories offer insights into computational problem complexity.
- Phase transitions in parameter space can serve as predictors for problem difficulty.
- This methodology enhances the understanding and prediction of computational effort for hard problems.