Tight Bounds for Noisy Computation of High-Influence Functions, Connectivity, and Threshold

2025-02-07

AI Summary

This paper investigates query complexity for fundamental computational problems under the noisy query model, where each query response is independently flipped with a fixed probability. For high-influence Boolean functions, graph connectivity testing, k-threshold, and counting problems, the authors combine noise analysis, information-theoretic lower bounds, influence theory, and randomized algorithm design to establish the first tight asymptotic bounds: (1) Boolean functions with total influence Ω(n) require Θ(n log n) queries; (2) graph connectivity testing requires Θ(n² log n) queries—the first exact characterization; (3) both k-threshold and counting problems achieve (1±o(1))-optimal query complexity. All results provide the first tight characterizations in this noisy query model, significantly advancing the theoretical understanding of robust computation under persistent noise.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In the noisy query model, the (binary) return value of every query (possibly repeated) is independently flipped with some fixed probability $p in (0, 1/2)$. In this paper, we obtain tight bounds on the noisy query complexity of several fundamental problems. Our first contribution is to show that any Boolean function with total influence $Omega(n)$ has noisy query complexity $Theta(nlog n)$. Previous works often focus on specific problems, and it is of great interest to have a characterization of noisy query complexity for general functions. Our result is the first noisy query complexity lower bound of this generality, beyond what was known for random Boolean functions [Reischuk and Schmeltz, FOCS 1991]. Our second contribution is to prove that Graph Connectivity has noisy query complexity $Theta(n^2 log n)$. In this problem, the goal is to determine whether an undirected graph is connected using noisy edge queries. While the upper bound can be achieved by a simple algorithm, no non-trivial lower bounds were known prior to this work. Last but not least, we determine the exact number of noisy queries (up to lower order terms) needed to solve the $k$-Threshold problem and the Counting problem. The $k$-Threshold problem asks to decide whether there are at least $k$ ones among $n$ bits, given noisy query access to the bits. We prove that $(1pm o(1)) frac{nlog (min{k,n-k+1}/delta)}{(1-2p)log frac{1-p}p}$ queries are both sufficient and necessary to achieve error probability $delta = o(1)$. Previously, such a result was only known when $min{k,n-k+1}=o(n)$ [Wang, Ghaddar, Zhu and Wang, arXiv 2024]. We also show a similar $(1pm o(1)) frac{nlog (min{k+1,n-k+1}/delta)}{(1-2p)log frac{1-p}p}$ bound for the Counting problem, where one needs to count the number of ones among $n$ bits given noisy query access and $k$ denotes the answer.

Problem

• Determine noisy query complexity for Boolean functions.
• Establish noisy query complexity for Graph Connectivity.
• Solve k-Threshold and Counting problems with noisy queries.

Innovation

• Tight bounds for noisy query complexity
• Graph Connectivity with noisy edge queries
• Exact noisy queries for k-Threshold problem