Tight query complexity bounds for learning graph partitions

TL;DR

This paper establishes tight bounds on the number of membership queries needed to learn graph partitions and components, improving previous bounds and introducing new oracles for efficient graph property learning.

Contribution

It provides tight query complexity bounds for learning graph partitions, introduces an oracle for counting components with fewer queries, and extends results to learning and verifying graphs with fewer queries.

Findings

Lower bound of (k-1)n - C(k,2) queries for learning graph components

Matching the query complexity of a known algorithm with new bounds

An oracle that learns the number of components with fewer queries

Abstract

Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $\Omega(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle, with access to which one can learn the number of components of $G$ in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of $\widetilde\Theta(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.