Tight query complexity bounds for learning graph partitions

Xizhi Liu, Sayan Mukherjee
Proceedings of Thirty Fifth Conference on Learning Theory, PMLR 178:167-181, 2022.

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 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 it.

Cite this Paper

BibTeX
@InProceedings{pmlr-v178-liu22a, title = {Tight query complexity bounds for learning graph partitions}, author = {Liu, Xizhi and Mukherjee, Sayan}, booktitle = {Proceedings of Thirty Fifth Conference on Learning Theory}, pages = {167--181}, year = {2022}, editor = {Loh, Po-Ling and Raginsky, Maxim}, volume = {178}, series = {Proceedings of Machine Learning Research}, month = {02--05 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v178/liu22a/liu22a.pdf}, url = {https://proceedings.mlr.press/v178/liu22a.html}, 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 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 it.} }
Endnote
%0 Conference Paper %T Tight query complexity bounds for learning graph partitions %A Xizhi Liu %A Sayan Mukherjee %B Proceedings of Thirty Fifth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2022 %E Po-Ling Loh %E Maxim Raginsky %F pmlr-v178-liu22a %I PMLR %P 167--181 %U https://proceedings.mlr.press/v178/liu22a.html %V 178 %X 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 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 it.
APA
Liu, X. & Mukherjee, S.. (2022). Tight query complexity bounds for learning graph partitions. Proceedings of Thirty Fifth Conference on Learning Theory, in Proceedings of Machine Learning Research 178:167-181 Available from https://proceedings.mlr.press/v178/liu22a.html.

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