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

TL;DR

This work analyzes the noisy query model where each queried bit is independently flipped with probability $p\in(0,1/2)$. It introduces a three-phase lower-bound framework that strengthens Feige's approach and applies it to three core problems: (i) high-influence Boolean functions, (ii) Graph Connectivity, and (iii) Threshold/Counting. For high-influence functions, any $f$ with $\mathsf{I}(f)\ge c n$ requires $\Omega(n\log n)$ noisy queries, with tight $O(n\log n)$ upper bounds by per-bit repetition; Graph Connectivity requires $\Theta(n^2\log n)$ queries, with a hard distribution based on Uniform Spanning Trees driving the lower bound; and for k-Threshold and Counting, the paper shows exact leading constants $(1\pm o(1)) \frac{n \log (\cdot)}{D_{\mathrm{KL}}(p\parallel1-p)}$ in all regimes, generalizing previous $k=o(n)$ results. The techniques combine a detailed three-phase hardness construction, precise posterior analyses, and biased-influence arguments to tightly pin down query complexity. The results deepen our understanding of noisy query complexity for broad function classes and give tight, applicable bounds for foundational graph and counting tasks in noisy environments.

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 $Ω(n)$ has noisy query complexity $Θ(n\log n)$. 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 $Θ(n^2 \log n)$. 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 $(1\pm o(1)) \frac{n\log (\min\{k,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ queries are both sufficient and necessary to achieve error probability $δ= o(1)$. 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 $(1\pm o(1)) \frac{n\log (\min\{k+1,n-k+1\}/δ)}{(1-2p)\log \frac{1-p}p}$ 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.

Theorems & Definitions (77)

• Theorem 1.1: Noisy query complexity of high-influence functions
• Theorem 1.2: Hardness of Graph Connectivity
• Theorem 1.3: Noisy query complexity of $k$-Threshold
• Theorem 1.4: Noisy query complexity of Counting
• Lemma 3.1
• proof
• Theorem 4.1: Noisy query complexity of high-influence functions
• Lemma 4.1
• proof
• Proposition 4.2