A Fast Binary Splitting Approach for Non-Adaptive Learning of Erd\H{o}s--R\'enyi Graphs

TL;DR

This paper introduces an efficient non-adaptive method for learning Erdős–Rényi graphs using a binary splitting approach, achieving near-optimal test complexity with improved decoding time.

Contribution

It extends the binary splitting technique to Erdős–Rényi graph learning, reducing decoding time while maintaining optimal test complexity.

Findings

Achieves $O(ar{k} ext{log} n)$ tests for graph recovery.

Decoding time is improved to $O(ar{k}^{1+ ext{delta}} ext{log} n)$.

High probability of successful edge set recovery.

Abstract

We study the problem of learning an unknown graph via group queries on node subsets, where each query reports whether at least one edge is present among the queried nodes. In general, learning arbitrary graphs with $n$ nodes and $k$ edges is hard in the non-adaptive setting, requiring $\Omega\big(\min\{k^2\log n,\,n^2\}\big)$ tests even when a small error probability is allowed. We focus on learning Erd\H{o}s--R\'enyi (ER) graphs $G\sim\mathrm{ER}(n,q)$ in the non-adaptive setting, where the expected number of edges is $\bar{k}=q\binom{n}{2}$, and we aim to design an efficient testing--decoding scheme achieving asymptotically vanishing error probability. Prior work (Li--Fresacher--Scarlett, NeurIPS 2019) presents a testing--decoding scheme that attains an order-optimal number of tests $O(\bar{k}\log n)$ but incurs $\Omega(n^2)$ decoding time, whereas their proposed sublinear-time algorithm incurs an extra $(\log \bar{k})(\log n)$ factor in the number of tests. We extend the binary splitting approach, recently developed for non-adaptive group testing, to the ER graph learning setting, and prove that the edge set can be recovered with high probability using $O(\bar{k}\log n)$ tests while attaining decoding time $O(\bar{k}^{1+\delta}\log n)$ for any fixed $\delta>0$.