We study the complexity of determining the edge connectivity of a simple graph with cut queries. We show that (i) there is a bounded-error randomized algorithm that computes edge connectivity with $O(n)$ cut queries, and (ii) there is a bounded-error quantum algorithm that computes edge connectivity with $\~O(\sqrt{n})$ cut queries. We prove these results using a new technique called "star contraction" to randomly contract edges of a graph while preserving non-trivial minimum cuts. In star contraction vertices randomly contract an edge incident on a small set of randomly chosen vertices. In contrast to the related 2-out contraction technique of Ghaffari, Nowicki, and Thorup [SODA'20], star contraction only contracts vertex-disjoint star subgraphs, which allows it to be efficiently implemented via cut queries. The $O(n)$ bound from item (i) was not known even for the simpler problem of connectivity, and improves the $O(n\log^3 n)$ bound by Rubinstein, Schramm, and Weinberg [ITCS'18]. The bound is tight under the reasonable conjecture that the randomized communication complexity of connectivity is $\Omega(n\log n)$, an open question since the seminal work of Babai, Frankl, and Simon [FOCS'86]. The bound also excludes using edge connectivity on simple graphs to prove a superlinear randomized query lower bound for minimizing a symmetric submodular function. Item (ii) gives a nearly-quadratic separation with the randomized complexity and addresses an open question of Lee, Santha, and Zhang [SODA'21]. The algorithm can also be viewed as making $\~O(\sqrt{n})$ matrix-vector multiplication queries to the adjacency matrix. Finally, we demonstrate the use of star contraction outside of the cut query setting by designing a one-pass semi-streaming algorithm for computing edge connectivity in the vertex arrival setting. This contrasts with the edge arrival setting where two passes are required.
翻译:我们用切开的查询来研究确定简单图的边缘连接的复杂程度。 我们显示 (i) 存在一个以 $(n) 削减查询来计算边缘连接的封闭性自动随机算法, 以及 (ii) 存在一个以 $(O) (sqrt{n) 削减查询来计算边缘连接的封闭性自动算法。 我们用一种叫“Star 收缩”的新技术来证明这些结果, 用一种叫“Star 收缩”的方法来随机将图的边缘连接起来, 同时保留非三角最小的最小切换。 在恒星收收收缩时, 在一组随机选择的平流上随机选择的边缘算法。 与相关的加法里、诺威基和索鲁普的2点收缩法相比, 恒收收收只通过切查询来高效地实施电离子( 美元) 将( 美元(n) 捆绑绑起来的图, 即使是更简单的连结, 也是为了更简单地(n) 直径(n) 直径) 和直径(ralalal) IM) 的离。