New Graph and Hypergraph Container Lemmas with Applications in Property Testing

TL;DR

This work extends the graph container framework to property testing by proving two new container lemmas: a hypergraph container lemma for satisfiability that yields a sample complexity of $\widetilde{O}(\frac{k q^3}{\epsilon})$ for $(q,k)$-SAT and a graph container lemma for independent set stars that achieves $\widetilde{O}(\frac{\rho^5}{\epsilon^{7/2}})$ query complexity for $\rho$-IndepSet. These tools unlock strong results for canonical testers (e.g., for satisfiability and colorability) and reveal a separation between canonical and non-canonical testers in non-homogeneous partition settings. The approach hinges on carefully designed fingerprint/container schemes with bounds on fingerprint size and container contents, enabling precise probabilistic analyses. The findings broaden the applicability of container methods in algorithmic property testing and improve understanding of sample vs. query complexities for key graph and hypergraph properties.

Abstract

The graph and hypergraph container methods are powerful tools with a wide range of applications across combinatorics. Recently, Blais and Seth (FOCS 2023) showed that the graph container method is particularly well-suited for the analysis of the natural canonical tester for two fundamental graph properties: having a large independent set and $k$-colorability. In this work, we show that the connection between the container method and property testing extends further along two different directions. First, we show that the container method can be used to analyze the canonical tester for many other properties of graphs and hypergraphs. We introduce a new hypergraph container lemma and use it to give an upper bound of $\widetilde{O}(kq^3/ε)$ on the sample complexity of $ε$-testing satisfiability, where $q$ is the number of variables per constraint and $k$ is the size of the alphabet. This is the first upper bound for the problem that is polynomial in all of $k$, $q$ and $1/ε$. As a corollary, we get new upper bounds on the sample complexity of the canonical testers for hypergraph colorability and for every semi-homogeneous graph partition property. Second, we show that the container method can also be used to study the query complexity of (non-canonical) graph property testers. This result is obtained by introducing a new container lemma for the class of all independent set stars, a strict superset of the class of all independent sets. We use this container lemma to give a new upper bound of $\widetilde{O}(ρ^5/ε^{7/2})$ on the query complexity of $ε$-testing the $ρ$-independent set property. This establishes for the first time the non-optimality of the canonical tester for a non-homogeneous graph partition property.

Figures (2)

• Figure 1: An illustration of the two hyperedges formed from the constraint that the variables $x_1$, $x_2$, and $x_3$ must not all be equal.
• Figure 2: An independent set star with a core of size $4$ (circled) and $6$ outer vertices. Dashed lines represent pairs of vertices that are not adjacent in the graph. The outer vertices may or may not be adjacent to each other.

Theorems & Definitions (30)

• Theorem 1
• Corollary 2
• Corollary 3
• Theorem 4
• Lemma 5: Lemma 5 in blaisSeth
• Lemma 6
• Lemma 7
• Remark
• Proposition 8
• proof