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 $\Omega((2m/d)^{d/2})$ 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 $\Delta$ and edge size ratio $\rho$, we give a non-adaptive algorithm with $O((2n)^{\rho \Delta+1}\log^2 n)$ 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.
翻译:我们研究如何通过边缘检测查询来学习高光学的问题。 在这个问题中, 学习者会询问隐藏高光学的顶端, 并观察这些顶端是否包含边缘。 一般来说, 学习最高大小为$d$的高光学需要$\Omega (( 2m/ d)\\\\ d/2}) 查询。 在本文中, 我们的目标是找出可以学习而不会因查询复杂性在边缘体积上成倍增长而痛苦的高光学家族。 我们显示, 高超和低度近于单面的顶端高光学子是否包含边际 。 一般来说, 学习最高尺寸为$( 2/ d) 美元( ) 的顶端高超光度高光学和低度的顶端高超光度高光学, 最高数值为$( =) 美元( log\\\ log) 最高值( =美元) 的顶级算算法, 我们给出$( =_ 美元( legal) legon) 最高高度的顶级算数。