Learning Graph Partitions

• URL: http://arxiv.org/abs/2112.07897v1
• Date: Wed, 15 Dec 2021 05:28:45 GMT
• Title: Learning Graph Partitions
• Authors: Sayan Mukherjee
• Abstract summary: 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 $nge kge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $frac12(n-k)(k-1)$ membership queries.
• Score: 2.3224617218247126
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\frac{1}{2}(n-k)(k-1)$ membership queries. This proves the optimality of the $O(nk)$ algorithm proposed by Reyzin and Srivastava (2007) for this problem, improving on the best known information-theoretic bound of $\Omega(n\log k)$ queries. Further, we construct an oracle that 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 this oracle.

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