Minimum $s$--$t$ Cuts with Fewer Cut Queries

Abstract

We study the problem of computing a minimum $s$--$t$ cut in an unweighted, undirected graph via \emph{cut queries}. In this model, the input graph is accessed through an oracle that, given a subset of vertices $S \subseteq V$, returns the size of the cut $(S, V \setminus S)$. This line of work was initiated by Rubinstein, Schramm, and Weinberg (ITCS 2018), who gave a randomized algorithm that computes a minimum $s$--$t$ cut using $\widetilde{O}(n^{5/3})$ queries, thereby showing that one can avoid spending $\widetilde{\Theta}(n^2)$ queries required to learn the entire graph. A recent result by Anand, Saranurak, and Wang (SODA 2025) also matched this upper bound via a deterministic algorithm based on blocking flows. In this work, we present a new randomized algorithm that improves the cut-query complexity to $\widetilde{O}(n^{8/5})$. At the heart of our approach is a query-efficient subroutine that incrementally reveals the graph edge-by-edge while increasing the maximum $s$--$t$ flow in the learned subgraph at a rate faster than classical augmenting-path methods. Notably, our algorithm is simple, purely combinatorial, and can be naturally interpreted as a recursive greedy procedure. As a further consequence, we obtain a \emph{deterministic} and \emph{combinatorial} two-party communication protocol for computing a minimum $s$--$t$ cut using $\widetilde{O}(n^{11/7})$ bits of communication. This improves upon the previous best bound of $\widetilde{O}(n^{5/3})$, which was obtained via reductions from the aforementioned cut-query algorithms. In parallel, it has been observed that an $\widetilde{O}(n^{3/2})$-bit randomized protocol can be achieved via continuous optimization techniques; however, these methods are fundamentally different from our combinatorial approach.