Non-Adaptive Learning of Sparse Erdős--Rényi Graphs via Affine Splitting

Hoang Ta

Graph learning from edge-detecting queries concerns the reconstruction of an unknown edge set on a known vertex set. Each query reports whether a specified vertex subset contains at least one edge. We study non-adaptive schemes, in which all queries are fixed before any outcomes are observed, with the goal of achieving exact recovery using few queries and fast decoding. For general graphs on $n$ vertices with at most $k$ edges, non-adaptive recovery requires $Ω(\min\{k^2\log n,n^2\})$ queries in the worst case, even when a small error probability is allowed. In this paper, we consider Erdős--Rényi ($\mathrm{ER}$) graphs $G\sim \mathrm{ER}(n,q)$, with expected edge count $\bar{k}=q\binom{n}{2}$. Our scheme uses $O(\bar{k}\log n)$ queries and achieves exact recovery in $O(\bar{k}\log n)$ decoding time with probability tending to one throughout the regime $\bar{k}\to\infty$ and $\bar{k}=o(n^2)$. This improves the previous $O(\bar{k}^{1+δ}\log n)$ decoding guarantee for any fixed $δ>0$, while maintaining the same query order. The guarantee also extends beyond the previously studied regime $\bar{k}=Θ(n^{2θ})$ with fixed $θ\in(0,1)$. Our approach builds on the binary splitting method…