• Corpus ID: 249848294

Tight query complexity bounds for learning graph partitions

@inproceedings{Liu2022TightQC,
  title={Tight query complexity bounds for learning graph partitions},
  author={Xizhi Liu and Sayan Mukherjee},
  booktitle={COLT},
  year={2022}
}

• Xizhi Liu, Sayan Mukherjee
• Published in COLT 15 December 2021
• Computer Science, Mathematics

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 ≥ k ≥ 2, learning the components of an n -vertex hidden graph with k components requires at least ( k − 1) n − (cid:0) k 2 (cid:1) membership queries. Our result improves on the best known information-theoretic bound of Ω( n log k ) queries, and exactly matches the query complexity of the algorithm introduced by Reyzin and Srivastava [22] 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 (cid:101) Θ( m ) queries for both learning and verifying an m -edge hidden graph G using it.