[Paper Review] On the Characterization of $1$-sided error Strongly-Testable Graph Properties for bounded-degree graphs, including an appendix

Hiro Ito, Areej Khoury+1 more|arXiv (Cornell University)|Sep 22, 2019

Complexity and Algorithms in Graphs21 references4 citations

TL;DR

This paper provides a characterization of 1-sided error strongly-testable monotone and hereditary graph properties in bounded-degree undirected and directed graphs. It shows that such properties are strongly-testable if and only if they are close to being defined by a finite set of forbidden subgraphs (for monotone) or induced subgraphs (for hereditary), with the distance measured in terms of edge edits relative to the total number of edges.

ABSTRACT

We study property testing of (di)graph properties in bounded-degree graph models. The study of graph properties in bounded-degree models is one of the focal directions of research in property testing in the last 15 years. However, despite of the many results and the extensive research effort, there is no characterization of the properties that are strongly-testable (i.e., testable with constant query complexity) even for $1$-sided error tests. The bounded-degree model can naturally be generalized to directed graphs resulting in two models that were considered in the literature. The first contains the directed graphs in which the outdegree is bounded but the indegree is not restricted. In the other, both the outdegree and indegree are bounded. We give a characterization of the $1$-sided error strongly-testable {\em monotone} graph properties, and the $1$-sided error strongly-testable {\em hereditary} graph properties in all the bounded-degree directed and undirected graphs models.

Motivation & Objective

• To address the long-standing open problem of characterizing 1-sided error strongly-testable graph properties in bounded-degree graph models.
• To extend existing results on property testing in bounded-degree graphs by focusing on monotone and hereditary properties with 1-sided error.
• To unify the understanding of testability across different bounded-degree models: undirected, F(d)-model (out-degree bounded), and FB(d)-model (in- and out-degree bounded).
• To establish a necessary and sufficient condition for strong testability using finite forbidden configurations, generalizing known results on minor-freeness and subgraph-freeness.
• To explore the limitations of extending the characterization beyond monotone and hereditary properties, particularly in cases where local corrections may introduce new violations.

Proposed method

• Define the bounded-degree graph model and distinguish between the F(d)-model (out-degree bounded) and FB(d)-model (in- and out-degree bounded), including their query access mechanisms.
• Introduce the concept of 'closeness' to a forbidden configuration set, where a graph is ε-close to a property if at most ε·dn edges need to be modified to satisfy the property.
• Use the notion of forbidden subgraphs (for monotone properties) and forbidden induced subgraphs (for hereditary properties) as the core structural component of the characterization.
• Apply a decomposition argument: for a graph ε-far from a k-star-free minor property, iteratively remove small connected components to reduce the graph to one with no large components, showing that only O(εdn) edges need to be removed.
• Prove that if a graph is ε-far from a property defined by a finite set of forbidden configurations, then it must contain many occurrences of these configurations, especially in low-degree models like FB(1).
• Leverage structural results from graph minor theory and local partitioning techniques to bound the number of edge edits required to make a graph satisfy the forbidden configuration condition.

Experimental results

Research questions

• RQ1 What characterizes 1-sided error strongly-testable monotone graph properties in bounded-degree undirected and directed graphs?
• RQ2 Can hereditary graph properties be similarly characterized in terms of forbidden induced subgraphs for 1-sided error testing?
• RQ3 Under what conditions is a graph ε-far from a property defined by a finite set of forbidden configurations guaranteed to contain many such configurations?
• RQ4 Why does the characterization fail to extend to general properties beyond monotone and hereditary ones, especially when local corrections may create new violations?
• RQ5 In the special case of the FB(1)-model (2-bounded degree), why does the inverse of the characterization hold, and what global structural properties enable this?

Key findings

• A monotone graph property is 1-sided error strongly-testable in all bounded-degree models if and only if it is ε-close to a property defined by a finite set of forbidden subgraphs of constant size.
• For hereditary properties, the characterization holds with forbidden induced subgraphs instead of subgraphs, providing a complete structural characterization.
• The property of being k-star-free as a minor is 1-sided error strongly-testable in the undirected d-bounded degree model, with the number of edge edits required bounded by ε·dn for any ε > 0.
• In the FB(1)-model (and thus the undirected 2-bounded degree model), if a graph is ε-far from being C-free, then it contains Ω(εn) occurrences of the forbidden configurations, establishing a strong global connection.
• The characterization fails to extend to general properties due to potential global conflicts where correcting one forbidden configuration introduces another, a phenomenon not present in monotone or hereditary settings.
• The results provide a foundational step toward a full characterization of all 1-sided error strongly-testable graph properties in bounded-degree models, particularly by identifying the role of finite forbidden configurations as the key structural invariant.