稀疏 Erdős--Rényi 图的非自适应学习：基于仿射分割

中文摘要 AI 辅助

基于边检测查询的图学习问题涉及在已知顶点集上重建未知边集。每次查询报告指定顶点子集是否包含至少一条边。我们研究非自适应方案，其中所有查询在观察到任何结果之前固定，目标是用少量查询和快速解码实现精确恢复。对于具有至多 k 条边的 n 个顶点的一般图，非自适应恢复在最坏情况下需要 Ω(min{k²log n, n²}) 次查询，即使允许较小的错误概率。在本文中，我们考虑 Erdős--Rényi (ER) 图 G∼ER(n,q)，期望边数为 k̄=q·C(n,2)。我们的方案使用 O(k̄ log n) 次查询，并在 k̄→∞ 且 k̄=o(n²) 的整个范围内以趋于 1 的概率在 O(k̄ log n) 解码时间内实现精确恢复。这改进了先前对任意固定 δ>0 的 O(k̄^{1+δ} log n) 解码保证，同时保持相同的查询阶。该保证还扩展到先前研究的 k̄=Θ(n^{2θ})（固定 θ∈(0,1)）范围之外。我们的方法建立在先前工作中使用的二分分割方法之上，该方法将顶点组织成逐次变小的组的层次结构。我们引入了三个主要变化：(i) 我们使用有限域上的随机仿射哈希函数以常数时间处理每个候选对；(ii) 我们将分割过程直接应用于完整图，避免组合多个较小图学习子问题的解；(iii) 我们直接限制总解码工作量，而不是在每个级别上推导候选数量的单独高概率界限。

英文摘要

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 used in prior work, which organizes vertices into a hierarchy of successively smaller groups. We introduce three main changes: (i) we use random affine hash functions over a finite field to process each candidate pair in constant time; (ii) we apply the splitting procedure directly to the full graph, avoiding the need to combine solutions to multiple smaller graph-learning subproblems; and (iii) we bound the total decoding workload directly rather than deriving separate high-probability bounds on candidate counts at each level.

发表机构

• Hanoi University of Science and Technology（河内理工大学）

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