Cut query algorithms with star contraction

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(√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(nlog3 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(nlog 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(√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.