A Tolerant Independent Set Tester

2025-03-27

AI Summary

This paper studies fault-tolerant testing of the ρ-independent set property in dense graphs: distinguishing between graphs containing a ρn-vertex induced subgraph with at most δ(ρn)² edges versus those where every ρn-vertex induced subgraph has at least (δ + ε)(ρn)² edges. We introduce the first graph container lemma applicable to sparse subgraphs—not merely independent sets—thereby overcoming fundamental limitations of prior container-based approaches. Leveraging this lemma, we design a fault-tolerant tester requiring only Õ(ρ³/ε²) uniformly random vertex samples. This sample complexity is nearly optimal, matching the state-of-the-art upper bound for non-fault-tolerant testing (up to logarithmic factors). Moreover, our work extends independent set counting techniques to sparse subgraph counting for the first time, yielding novel combinatorial generalizations and advancing the theoretical foundations of property testing in dense graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

Abstract

We give nearly optimal bounds on the sample complexity of $(widetilde{Omega}(epsilon),epsilon)$-tolerant testing the $ ho$-independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random subgraph on $widetilde{O}( ho^3/epsilon^2)$ vertices and, for some constant $c,$ distinguishes between graphs that have an induced subgraph of size $ ho n$ with fewer than $frac{epsilon}{c log^4(1/epsilon)} n^2$ edges from graphs for which every induced subgraph of size $ ho n$ has at least $epsilon n^2$ edges. Our sample complexity bound matches, up to logarithmic factors, the recent upper bound by Blais and Seth (2023) for the non-tolerant testing problem, which is known to be optimal for the non-tolerant testing problem based on a lower bound by Feige, Langberg and Schechtman (2004). Our main technique is a new graph container lemma for sparse subgraphs instead of independent sets. We also show that our new lemma can be used to generalize one of the classic applications of the container method, that of counting independent sets in regular graphs, to counting sparse subgraphs in regular graphs.

Problem

• Tolerant testing of ρ-independent set property in dense graphs
• Sample complexity bounds for distinguishing sparse subgraphs
• Generalizing graph container lemma for sparse subgraphs

Innovation

• Tolerant independent set testing algorithm
• Graph container lemma for sparse subgraphs
• Generalizes counting independent sets method