Learning Low Degree Hypergraphs

TL;DR

The paper addresses learning hidden hypergraphs under edge-detecting queries, showing that while general hypergraphs demand exponential query effort, certain families admit poly$(n,m)$ query learning. It introduces unique-edge covering families as a key tool and develops two edge-finding subroutines to construct a scalable learning framework, enabling poly-query learning for hypermatchings ($oldsymbol{ riangle=1}$) and low-degree near-uniform hypergraphs. The main contributions include an $oldsymbol{ ilde{O}}( ext{log}^3 n)$-round algorithm with $oldsymbol{O}(n ext{log}^5 n)$ queries for hypermatchings, a lower bound of $oldsymbol{ ilde{Ω}}( ext{log log } n)$ rounds for poly-query learning, and a non-adaptive algorithm with $oldsymbol{O}((2n)^{ ho riangle+1} ext{log}^2 n)$ queries for general $oldsymbol{ riangle}$ and $oldsymbol{ ho}$, complemented by hardness results for non-adaptive and o$( ext{log log } n)$-round settings. These results constitute the first poly$(n,m)$-level algorithms for learning non-trivial hypergraph families with super-constant edges, with substantial implications for combinatorial learning and related applications. The techniques center on unique-edge coverings and adaptive vs. non-adaptive strategies, offering a principled path to efficiently identify reaction- or interaction-structure in complex systems modeled by hypergraphs.

Abstract

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.

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