学习低度超图
Learning Low Degree Hypergraphs
AI总结：
研究通过边检测查询学习超图的问题，针对超匹配和低度近均匀超图提出了多项式查询复杂度的自适应与非自适应算法，突破了指数级查询复杂度限制。
摘要：
我们研究通过边检测查询学习超图的问题。在此问题中，学习者查询隐藏超图的顶点子集，并观察这些子集是否包含边。通常，学习具有 $m$ 条最大大小为 $d$ 的边的超图需要 $Ω((2m/d)^{d/2})$ 次查询。在本文中，我们旨在识别那些可以在不遭受随边大小呈指数级增长的查询复杂度的情况下学习的超图族。我们证明具有 $n$ 个顶点的超匹配和低度近均匀超图可以用 poly$(n)$ 次查询学习。对于学习超匹配（最大度为 $ 1$ 的超图），我们给出一个 $O(\log^3 n)$ 轮算法，使用 $O(n \log^5 n)$ 次查询。我们通过证明不存在在 $o(\log \log n)$ 自适应轮内学习超匹配的 poly$(n)$ 查询算法来补充此上界。对于具有最大度 $Δ$ 和边大小比 $ρ$ 的超图，我们给出一个使用 $O((2n)^{ρΔ+1}\log^2 n)$ 次查询的非自适应算法。据我们所知，这些是首批具有 poly$(n, m)$ 查询复杂度、用于学习具有超常数数量和超常数大小边的非平凡超图族的算法。
英文摘要：
We study the problem of learning a hypergraph via edge detecting queries. In this problem, a learner queries subsets of vertices of a hidden hypergraph and observes whether these subsets contain an edge or not. In general, learning a hypergraph with $m$ edges of maximum size $d$ requires $Ω((2m/d)^{d/2})$ queries. In this paper, we aim to identify families of hypergraphs that can be learned without suffering from a query complexity that grows exponentially in the size of the edges. We show that hypermatchings and low-degree near-uniform hypergraphs with $n$ vertices are learnable with poly$(n)$ queries. For learning hypermatchings (hypergraphs of maximum degree $ 1$), we give an $O(\log^3 n)$-round algorithm with $O(n \log^5 n)$ queries. We complement this upper bound by showing that there are no algorithms with poly$(n)$ queries that learn hypermatchings in $o(\log \log n)$ adaptive rounds. For hypergraphs with maximum degree $Δ$ and edge size ratio $ρ$, we give a non-adaptive algorithm with $O((2n)^{ρΔ+1}\log^2 n)$ queries. To the best of our knowledge, these are the first algorithms with poly$(n, m)$ query complexity for learning non-trivial families of hypergraphs that have a super-constant number of edges of super-constant size.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
