New Graph and Hypergraph Container Lemmas with Applications in Property Testing
arxiv(2024)
摘要
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
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
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.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要