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

TL;DR

This paper introduces a non-adaptive algorithm for estimating edges and sampling in graphs using bipartite independent set queries, achieving significant improvements over prior adaptive methods in query complexity and round efficiency.

Contribution

It presents the first non-adaptive algorithms for edge estimation and sampling in this model, with improved query complexities and fewer rounds than previous adaptive algorithms.

Findings

Non-adaptive edge estimation with $ ilde O( ext{poly}(1/ extepsilon), ext{poly}( extlog n))$ queries.

Non-adaptive nearly uniform edge sampling with $ ilde O( extepsilon^{-6} extlog^6 n)$ queries.

Connectivity algorithm with $ ilde O(n extlog^8 n)$ queries and two rounds of adaptivity.

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 $\Omega(\log^2 n)$ 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 $\Omega(\log^3 n)$ 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 $o(n^2)$ queries.