Improved Approximation Algorithms for Optimization Problems in Graphs with Superlogarithmic Treewidth 1st edition by Artur Czumaj, Andrzej Lingas, Johan Nilsson ISBN 3540206958 9783540206958
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Authors:Artur Czumaj, Andrzej Lingas; Johan Nilsson , Tags:Algorithms and Computation , Author sort:Artur Czumaj, Andrzej Lingas & Nilsson, Johan , Languages:Languages:eng , Published:Published:Oct 2003
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ISBN 10: 3540206958
ISBN 13: 978-3540206958
Author: Artur Czumaj, Andrzej Lingas, Johan Nilsson
-hard graph problems constrained to partial k-trees. Our first scheme yields deterministic polynomial-time algorithms achieving typically an approximation factor of k/log1 − ε n, where k = polylog (n). The second scheme yields randomized polynomial-time algorithms achieving an approximation factor of k / log n for k = ω(log n). Both our approximation methods lead to the best known approximation guarantees for some basic optimization problems. In particular, we obtain best known polynomial-time approximation guarantees for the classical maximum independent set problem in partial trees.
Improved Approximation Algorithms for Optimization Problems in Graphs with Superlogarithmic Treewidth 1st Table of contents:
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Introduction
- 1.1 Background and Motivation
- 1.2 Importance of Treewidth in Graph Theory
- 1.3 Overview of Approximation Algorithms
- 1.4 Main Contributions
- 1.5 Structure of the Paper
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Preliminaries
- 2.1 Graph Theory Basics
- 2.1.1 Definitions and Notations
- 2.1.2 Optimization Problems in Graphs
- 2.2 Treewidth and its Role in Graph Problems
- 2.2.1 Definition of Treewidth
- 2.2.2 Superlogarithmic Treewidth and its Significance
- 2.3 Approximation Algorithms
- 2.3.1 Approximation Ratio
- 2.3.2 Techniques for Designing Approximation Algorithms
- 2.1 Graph Theory Basics
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Related Work
- 3.1 Approximation Algorithms for Graph Problems with Bounded Treewidth
- 3.2 Algorithms for Graphs with Superlogarithmic Treewidth
- 3.3 Treewidth and its Implications for Approximation Complexity
- 3.4 Existing Methods and Their Limitations
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Optimization Problems in Graphs with Superlogarithmic Treewidth
- 4.1 Key Graph Problems Addressed in the Paper
- 4.1.1 Vertex Cover
- 4.1.2 Independent Set
- 4.1.3 Maximum Matching
- 4.1.4 Steiner Tree
- 4.2 Characterizing Graphs with Superlogarithmic Treewidth
- 4.2.1 Structural Properties
- 4.2.2 Relationship with Other Graph Classes
- 4.1 Key Graph Problems Addressed in the Paper
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Improved Approximation Algorithms
- 5.1 Overview of the Algorithmic Approach
- 5.2 Algorithm for Vertex Cover in Graphs with Superlogarithmic Treewidth
- 5.2.1 Algorithm Design
- 5.2.2 Approximation Guarantees
- 5.2.3 Experimental Results
- 5.3 Algorithm for Independent Set in Graphs with Superlogarithmic Treewidth
- 5.3.1 Algorithm Design
- 5.3.2 Approximation Guarantees
- 5.3.3 Experimental Results
- 5.4 Algorithm for Maximum Matching in Graphs with Superlogarithmic Treewidth
- 5.4.1 Algorithm Design
- 5.4.2 Approximation Guarantees
- 5.4.3 Experimental Results
- 5.5 Algorithm for Steiner Tree in Graphs with Superlogarithmic Treewidth
- 5.5.1 Algorithm Design
- 5.5.2 Approximation Guarantees
- 5.5.3 Experimental Results
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Analysis of Approximation Guarantees
- 6.1 Theoretical Analysis of Approximation Factors
- 6.2 Comparative Analysis of New Algorithms with Existing Approaches
- 6.3 Scalability and Efficiency of the Proposed Algorithms
- 6.4 Sensitivity to Treewidth and Other Graph Parameters
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Complexity of Graph Problems with Superlogarithmic Treewidth
- 7.1 Exact Algorithms for Graphs with Superlogarithmic Treewidth
- 7.2 Approximation Boundaries in Graphs with Superlogarithmic Treewidth
- 7.3 Computational Hardness of Optimization Problems
- 7.4 Connections to Other Complexity Classes (e.g., FPT, NP-Hard)
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Applications and Implications
- 8.1 Real-World Graphs with Superlogarithmic Treewidth
- 8.1.1 Network Design Problems
- 8.1.2 Computational Biology and Molecular Networks
- 8.1.3 Social Network Analysis
- 8.2 Impact of Improved Approximation on Practical Applications
- 8.3 Applications in Approximation for Large-Scale Graphs
- 8.1 Real-World Graphs with Superlogarithmic Treewidth
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Open Problems and Future Work
- 9.1 Generalization of Results to Other Graph Classes
- 9.2 Improvements in Approximation Algorithms
- 9.3 Combining Approximation with Exact Algorithms
- 9.4 Extending to Dynamic Graphs and Streaming Algorithms
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Conclusion
- 10.1 Summary of Contributions
- 10.2 Final Thoughts on Approximation Algorithms for Superlogarithmic Treewidth
- 10.3 Future Directions in Graph Optimization Problems
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