A Generalized Gale Shapley Algorithm for a Discrete Concave Stable Marriage Model 1st edition by Akinobu Eguchi, Satoru Fujishige, Akihisa Tamura ISBN 3540206958 9783540206958

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ISBN 10:       3540206958
ISBN 13: 978-3540206958
Author:         Akinobu Eguchi, Satoru Fujishige, Akihisa Tamura
 

A Generalized Gale-Shapley Algorithm for a Discrete-Concave Stable-Marriage Model 1st Table of contents:

1. Introduction

   • 1.1 Overview of the Stable Marriage Problem
   • 1.2 The Gale-Shapley Algorithm and Its Importance
   • 1.3 Motivation for Extending the Gale-Shapley Algorithm
   • 1.4 The Discrete-Concave Stable-Marriage Model
   • 1.5 Contribution of the Paper
   • 1.6 Structure of the Paper

2. Background and Related Work

   • 2.1 Classical Stable Marriage Problem
   • 2.2 The Gale-Shapley Algorithm and Its Solution Properties
   • 2.3 Extensions and Generalizations of Stable Marriage Models
   • 2.4 Discrete-Concave Functions and Their Role in Stability
   • 2.5 Previous Work on Discrete-Concave Models and Algorithmic Approaches
   • 2.6 Gaps and Challenges in Existing Approaches

3. Problem Formulation

   • 3.1 Formal Definition of the Stable Marriage Problem with Discrete-Concave Preferences
   • 3.2 Stability Criteria for the Discrete-Concave Model
   • 3.3 Key Differences Between Classical and Discrete-Concave Models
   • 3.4 Assumptions and Constraints in the Problem Setup
   • 3.5 Motivation for a Generalized Gale-Shapley Algorithm

4. Generalized Gale-Shapley Algorithm

   • 4.1 Overview of the Generalized Approach
   • 4.2 Key Modifications to the Classic Gale-Shapley Algorithm
   • 4.3 Step-by-Step Explanation of the Generalized Algorithm
   • 4.4 Handling Discrete-Concave Preferences in the Algorithm
   • 4.5 Convergence and Termination Properties
   • 4.6 Computational Complexity Analysis

5. Theoretical Analysis

   • 5.1 Existence of Stable Matchings in the Discrete-Concave Model
   • 5.2 Optimality and Stability of the Generalized Algorithm
   • 5.3 Comparison with Other Matching Algorithms (e.g., Deferred Acceptance, Top Trading Cycles)
   • 5.4 Proof of Correctness and Convergence
   • 5.5 Computational Complexity of the Generalized Gale-Shapley Algorithm

6. Performance Evaluation

   • 6.1 Experimental Setup and Methodology
   • 6.2 Benchmarking the Generalized Algorithm Against Classical Gale-Shapley
   • 6.3 Case Studies and Simulation Results
   • 6.4 Evaluation of Efficiency and Convergence Rates
   • 6.5 Comparison with Other Discrete-Concave Matching Approaches
   • 6.6 Impact of Discrete-Concave Preferences on Matching Outcomes

7. Applications

   • 7.1 Applications in Economics: Matching Markets and Resource Allocation
   • 7.2 Applications in Job Matching and College Admissions
   • 7.3 Applications in Computational Biology: Gene Matching
   • 7.4 Applications in Social Networks and Online Platforms
   • 7.5 Other Practical Use Cases in Allocation Problems

8. Challenges and Limitations

   • 8.1 Scalability of the Generalized Gale-Shapley Algorithm
   • 8.2 Handling Large-Scale Matching Problems with Discrete-Concave Preferences
   • 8.3 Limitations in Real-World Implementations of the Model
   • 8.4 Trade-Offs Between Algorithmic Complexity and Solution Quality
   • 8.5 Robustness of the Algorithm in Non-Standard Environments

9. Future Directions

   • 9.1 Extending the Generalized Gale-Shapley Algorithm to Multi-Agent Settings
   • 9.2 Generalizing to Multi-Objective Stable Marriage Models
   • 9.3 Incorporating Fuzzy or Uncertain Preferences in the Matching Model
   • 9.4 Hybrid Algorithms Combining Gale-Shapley with Other Matching Techniques
   • 9.5 Future Research on Discrete-Concave Models in Other Optimization Problems

10. Conclusion

• 10.1 Summary of Key Contributions and Findings
• 10.2 Impact of the Generalized Gale-Shapley Algorithm on Stable Marriage Problems
• 10.3 Insights into Discrete-Concave Stability Models
• 10.4 Final Remarks on the Evolution of Matching Algorithms and Their Applications

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