Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Abstract: We study the problem of estimating the number of edges in an $n$-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (ITCS '18). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a $(1\pm \epsilon)$ relative error approximation to the number of edges, with query complexity $$\tilde O(\epsilon^{{-5}\log^{5}} n )$$, where $\tilde O(\cdot)$ hides $\textrm{poly}(\log \log n)$ dependencies. This is the first non-adaptive algorithm in this setting achieving $\textrm{poly}(1/\epsilon,\log n)$ query complexity. Prior work requires $\Omega(\log² n)$ rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant $\epsilon$, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires $O(\epsilon^{-2} \log^{11} n)$ queries. Building on our edge estimation result, we give the first non-adaptive algorithm for outputting a nearly uniformly sampled edge with query complexity $$\tilde O(\epsilon^{-6} \log^{6} n)$$, improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require $\Omega(\log³ n)$ rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a $\tilde O(n\log^ 8 n)$ query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making $o(n<sup>2)$ queries.