Algorithms for Enumerating Circuits in Matroids 1st edition by Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Leonid Khachiyan ISBN 3540206958 9783540206958

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Product details:

ISBN 10:       3540206958
ISBN 13: 978-3540206958
Author:      Endre Boros, Khaled Elbassioni, Vladimir Gurvich, Leonid Khachiyan   
 

Algorithms for Enumerating Circuits in Matroids 1st Table of contents:

1. Introduction

   • 1.1 Motivation and Importance of Circuit Enumeration in Matroids
   • 1.2 Overview of Matroids and Their Key Properties
   • 1.3 Problem Definition: Enumerating Circuits in Matroids
   • 1.4 Contributions of the Paper
   • 1.5 Structure of the Paper

2. Background and Related Work

   • 2.1 Introduction to Matroid Theory
   • 2.2 Circuits in Matroids: Definitions and Properties
   • 2.3 The Role of Circuit Enumeration in Combinatorial Optimization
   • 2.4 Previous Work on Circuit Enumeration Algorithms
   • 2.5 Comparison of Approaches to Enumerating Circuits

3. Preliminaries

   • 3.1 Basic Definitions in Matroid Theory
   • 3.2 Representations of Matroids: Combinatorial and Geometric Views
   • 3.3 Fundamental Theorems in Matroid Theory
   • 3.4 Circuit Properties and Characterization
   • 3.5 Algorithmic Framework for Circuit Enumeration

4. Circuit Enumeration Algorithms

   • 4.1 Overview of the Algorithmic Approach
   • 4.2 Depth-First Search (DFS) Based Algorithm for Circuit Enumeration
   • 4.3 Efficient Backtracking Techniques for Circuit Enumeration
   • 4.4 Greedy Algorithms for Enumerating Specific Classes of Circuits
   • 4.5 Parallel and Distributed Algorithms for Circuit Enumeration
   • 4.6 Improvements in Space and Time Complexity for Circuit Enumeration
   • 4.7 Pseudo-code and Step-by-Step Explanation of the Algorithms

5. Complexity Analysis

   • 5.1 Time Complexity of Circuit Enumeration Algorithms
   • 5.2 Space Complexity Considerations
   • 5.3 Worst-Case and Average-Case Analysis
   • 5.4 Comparison with Other Enumerative Algorithms in Graph Theory
   • 5.5 Lower Bounds for Circuit Enumeration in Matroids

6. Applications of Circuit Enumeration

   • 6.1 Applications in Graph Theory: Cycle Enumeration and Connectivity
   • 6.2 Optimization Problems: Solving Maximum Flow and Network Design
   • 6.3 Applications in Coding Theory and Error Correction
   • 6.4 Circuit Enumeration in Game Theory and Economics
   • 6.5 Use in Machine Learning and Data Analysis

7. Experimental Results

   • 7.1 Experimental Setup and Methodology
   • 7.2 Benchmarking the Proposed Algorithms Against Existing Techniques
   • 7.3 Performance Metrics: Time, Space, and Correctness
   • 7.4 Results on Synthetic and Real-World Matroid Instances
   • 7.5 Discussion of Findings and Practical Insights

8. Challenges and Limitations

   • 8.1 Handling Large Matroids with Many Circuits
   • 8.2 Scalability Issues in High-Dimensional Matroids
   • 8.3 Efficiently Handling Special Classes of Matroids
   • 8.4 Dealing with Redundancy in the Enumeration Process
   • 8.5 Open Problems and Future Directions in Matroid Circuit Enumeration

9. Future Directions

   • 9.1 Improving Efficiency for Large-Scale Circuit Enumeration
   • 9.2 Extensions to Enumerating Other Combinatorial Structures
   • 9.3 Applications in Advanced Graph and Network Theory
   • 9.4 Integration with Other Optimization Algorithms (e.g., Linear Programming)
   • 9.5 Exploration of Quantum Algorithms for Circuit Enumeration

10. Conclusion

• 10.1 Summary of Key Contributions
• 10.2 Practical Impact of Efficient Circuit Enumeration Algorithms
• 10.3 Final Remarks and Open Problems

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