A Generalized Gale-Shapley Algorithm for a Discrete-Concave Stable-Marriage Model 1st edition by Akinobu Eguchi, Satoru Fujishige, Akihisa Tamura – Ebook PDF Instant Download/Delivery. 3540206958, 978-3540206958
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ISBN 10: 3540206958
ISBN 13: 978-3540206958
Author: Akinobu Eguchi, Satoru Fujishige, Akihisa Tamura
The stable marriage model due to Gale and Shapley is one of the most fundamental two-sided matching models. Recently, Fleiner generalized the model in terms of matroids, and Eguchi and Fujishige extended the matroidal model to the framework of discrete convex analysis. In this paper, we extend their model to a vector version in which indifference on preferences is allowed, and show the existence of a stable solution by a generalization of the Gale-Shapley algorithm.
A Generalized Gale-Shapley Algorithm for a Discrete-Concave Stable-Marriage Model 1st Table of contents:
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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