Bipartite Independent Set Oracles and Beyond: Can it Even Count Triangles in Polylogarithmic Queries?
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Beame et al. [ITCS 2018] introduced and used the Bipartite Independent Set (BIS) and Independent Set (IS) oracle access to an unknown, simple, unweighted and undirected graph and solved the edge estimation problem. The introduction of this oracle set forth a series of works in a short span of time that either solved open questions mentioned by Beame et al. or were generalizations of their work as in Dell and Lapinskas [STOC 2018], Dell, Lapinskas and Meeks [SODA 2020], Bhattacharya et al. [ISAAC 2019 and arXiv 2019], Chen et al. [SODA 2020]. Edge estimation using BIS can be done using polylogarithmic queries, while IS queries need sub-linear but more than polylogarithmic queries. Chen et al. improved Beame et al.'s upper bound result for edge estimation using IS and also showed an almost matching lower bound. This result was significant because this lower bound result on was the first lower bound result for independent set based oracles; till date no lower bound results exist for BIS. On the other hand, Beame et al. in their introductory work asked a few open questions out of which one was if structures of higher order than edges can be estimated using polylogarithmic number of BIS queries. Motivated by this question, we resolve in the negative by showing a lower bound (greater than polylogarithmic) for estimating the number of triangles using BIS. While doing so, we prove the first lower bound result involving BIS. We also provide a matching upper bound. Till now, query oracles were used for commensurate jobs – and for edge estimation, for triangle estimation, for hyperedge estimation. Ours is a work that uses a lower order oracle access, like to estimate a higher order structure like triangle.
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