Graph Verification with a Betweenness Oracle
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Proceedings of the 28th International Conference on Algorithmic Learning Theory, PMLR 76:238249, 2017.
Abstract
In this paper, we examine the query complexity of verifying a hidden graph $G$ with a betweenness oracle. Let $G=(V,E)$ be a hidden graph and $\hat{G}=(V,\hat{E})$ be a known graph. $V$ and $\hat{E}$ are known and $E$ is not known. The graphs are connected, unweighted and have bounded maximum degree $\Delta$. The task of the graph verification problem is to verify that $E=\hat{E}$. We have access to $G$ through a blackbox betweenness oracle. A betweenness oracle returns whether a vertex lies along a shortest path between two other vertices. The betweenness oracle nicely captures many realworld problems. We prove that graph verification can be done using $n^{1+o(1)}$ betweenness queries. Surprisingly, this matches the state of the art for the graph verification problem with the much stronger distance oracle. We also prove that graph verification requires $\Omega(n)$ betweenness queries  a matching lower bound.
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