Book contents
- Frontmatter
- Contents
- Preface
- Organization and Chapter Summaries
- Notation
- Acknowledgments
- 1 The Main Themes: Approximate Decision and Sublinear Complexity
- 2 Testing Linearity (Group Homomorphism)
- 3 Low-Degree Tests
- 4 Testing Monotonicity
- 5 Testing Dictatorships, Juntas, and Monomials
- 6 Testing by Implicit Sampling
- 7 Lower Bounds Techniques
- 8 Testing Graph Properties in the Dense Graph Model
- 9 Testing Graph Properties in the Bounded-Degree Graph Model
- 10 Testing Graph Properties in the General Graph Model
- 11 Testing Properties of Distributions
- 12 Ramifications and Related Topics
- 13 Locally Testable Codes and Proofs
- Appendix A Probabilistic Preliminaries
- Appendix B A Mini-Compendium of General Results
- Appendix C An Index of Specific Results
- References
- Index
8 - Testing Graph Properties in the Dense Graph Model
Published online by Cambridge University Press: 13 November 2017
- Frontmatter
- Contents
- Preface
- Organization and Chapter Summaries
- Notation
- Acknowledgments
- 1 The Main Themes: Approximate Decision and Sublinear Complexity
- 2 Testing Linearity (Group Homomorphism)
- 3 Low-Degree Tests
- 4 Testing Monotonicity
- 5 Testing Dictatorships, Juntas, and Monomials
- 6 Testing by Implicit Sampling
- 7 Lower Bounds Techniques
- 8 Testing Graph Properties in the Dense Graph Model
- 9 Testing Graph Properties in the Bounded-Degree Graph Model
- 10 Testing Graph Properties in the General Graph Model
- 11 Testing Properties of Distributions
- 12 Ramifications and Related Topics
- 13 Locally Testable Codes and Proofs
- Appendix A Probabilistic Preliminaries
- Appendix B A Mini-Compendium of General Results
- Appendix C An Index of Specific Results
- References
- Index
Summary
Summary: Following a general introduction to testing graph properties, this chapter focuses on the dense graph model, where graphs are represented by their adjacency matrix (predicate). The highlights of this chapter include:
1. A presentation of a natural class of graph properties that can each be tested within query complexity that is polynomial in the reciprocal of the proximity parameter. This class, called general graph partition problems, contains properties such as t-Colorability (for any t ≥ 2) and properties that refer to the density of the max-clique and to the density of the max-cut in a graph.
2. An exposition of the connection of testing (in this model) to Szemeredi's Regularity Lemma. The starting point and pivot of this exposition is the existence of constant-query (one-sided error) proximity-oblivious testers for all subgraph freeness properties.
We conclude this chapter with a taxonomy of known testers, organized according to their query complexity.
The current chapter is based on many sources; see Section 8.6.1 for details.
Organization. The current chapter is the first of a series of three chapters that cover three models for testing graph properties. In each model, we spell out the definition of property testing (when specialized to that model), present some of the known results, and demonstrate some of the ideas involved in the construction of testers (by focusing on testing Bipartiteness, which seems a good benchmark).
We start the current chapter with a general introduction to testing graph properties, which includes an overview of the three models (see Section 8.1.2). We then present and illustrate the “dense graph model” (Section 8.2), which is the focus of the current chapter. The main two sections (i.e., Sections 8.3 and 8.4) cover the two topics that are mentioned in the foregoing summary: Section 8.3 deals with testing arbitrary graph partition properties, as illustrated by the example of testing Bipartitness. Section 8.4 deals with the connection between property testing in this model and Szemeredi's Regularity Lemma, as illustrated by testing subgraph-freeness. The last two sections (i.e., Sections 8.5 and 8.6) are descriptive in nature: Section 8.5 presents a taxonomy of the known results, whereas Section 8.6 presents final comments.
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- Introduction to Property Testing , pp. 162 - 212Publisher: Cambridge University PressPrint publication year: 2017
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