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A Tolerant Independent Set Tester

Cameron Seth

TL;DR

This work addresses tolerant testing of the $\rho$-IndepSet property in dense graphs, achieving a near-optimal sample complexity of $\widetilde{O}(\rho^3/\epsilon^2)$ with a tolerant gap $(\epsilon/(c\log^4(1/\epsilon)),\epsilon)$. The main technical advance is a graph container lemma for sparse subgraphs, built from a fingerprint/revision framework, enabling containment guarantees even when induced subgraphs are only sparsely connected. The authors leverage this lemma to design a tolerant tester that matches the non-tolerant upper bound up to logarithmic factors and to generalize counting arguments for independent sets to counting sparse subgraphs in regular graphs. Together, these results push the frontier of tolerant graph property testing in the dense setting and open pathways to broader applications of sparse-subgraph containers, including counting problems in regular graphs.

Abstract

We give nearly optimal bounds on the sample complexity of $(\widetildeΩ(ε),ε)$-tolerant testing the $ρ$-independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random subgraph on $\widetilde{O}(ρ^3/ε^2)$ vertices and, for some constant $c,$ distinguishes between graphs that have an induced subgraph of size $ρn$ with fewer than $\fracε{c \log^4(1/ε)} n^2$ edges from graphs for which every induced subgraph of size $ρn$ has at least $εn^2$ 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.

A Tolerant Independent Set Tester

TL;DR

This work addresses tolerant testing of the -IndepSet property in dense graphs, achieving a near-optimal sample complexity of with a tolerant gap . The main technical advance is a graph container lemma for sparse subgraphs, built from a fingerprint/revision framework, enabling containment guarantees even when induced subgraphs are only sparsely connected. The authors leverage this lemma to design a tolerant tester that matches the non-tolerant upper bound up to logarithmic factors and to generalize counting arguments for independent sets to counting sparse subgraphs in regular graphs. Together, these results push the frontier of tolerant graph property testing in the dense setting and open pathways to broader applications of sparse-subgraph containers, including counting problems in regular graphs.

Abstract

We give nearly optimal bounds on the sample complexity of -tolerant testing the -independent set property in the dense graph setting. In particular, we give an algorithm that inspects a random subgraph on vertices and, for some constant distinguishes between graphs that have an induced subgraph of size with fewer than edges from graphs for which every induced subgraph of size has at least 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.

Paper Structure

This paper contains 16 sections, 10 theorems, 46 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Tolerant Independent Set Tester
  3. Main Technique: New Container Lemma
  4. Counting Sparse Subgraphs in Regular Graphs
  5. Preliminaries
  6. Notations.
  7. Chernoff Bound.
  8. A Tolerant Tester for the Independent Set Property
  9. Graph Container Lemma for Sparse Subgraphs
  10. Proof Overview
  11. Selecting a Revision Vertex.
  12. Fingerprint and Container Procedures
  13. An Initial Containment Lemma
  14. Revision Lemma
  15. Proof of Graph Container Lemma
  16. ...and 1 more sections

Key Result

Theorem 1

There exists a constant $c$ such that for any $\epsilon>0$ the sample complexity of $\left(\frac{\epsilon}{c \log^4(1/\epsilon)},\epsilon\right)$-tolerant testing the $\rho\textnormal{-IndepSet}\xspace$ property is $\widetilde{O}(\rho^3/\epsilon^2).$

Theorems & Definitions (21)

  • Theorem 1
  • Definition 2
  • Lemma 2
  • Theorem 3
  • Lemma 4: Chernoff's Bound
  • Theorem 4
  • proof
  • Remark 5
  • Definition 6
  • Lemma 7
  • ...and 11 more