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.

翻译：我们研究在无权无向图中通过\emph{割查询}计算最小 $s$--$t$ 割的问题。在此模型中，输入图通过一个预言机访问，给定顶点子集 $S \subseteq V$，该预言机返回割 $(S, V \setminus S)$ 的大小。这项研究由 Rubinstein、Schramm 和 Weinberg（ITCS 2018）开创，他们提出了一种随机算法，使用 $\widetilde{O}(n^{5/3})$ 次查询即可计算最小 $s$--$t$ 割，从而表明可以避免学习整个图所需的 $\widetilde{\Theta}(n^2)$ 次查询。Anand、Saranurak 和 Wang（SODA 2025）最近的一项成果也通过基于阻塞流的确定性算法达到了这一上界。在本工作中，我们提出了一种新的随机算法，将割查询复杂度改进为 $\widetilde{O}(n^{8/5})$。我们方法的核心是一个查询高效的子程序，它能够以比经典增广路径方法更快的速率，在逐步揭示图结构（每次一条边）的同时，增加已学习子图中的最大 $s$--$t$ 流。值得注意的是，我们的算法简单、纯粹组合，并且可以自然地解释为一个递归贪心过程。作为进一步的推论，我们获得了一个用于计算最小 $s$--$t$ 割的\emph{确定性}且\emph{组合的}两方通信协议，其通信复杂度为 $\widetilde{O}(n^{11/7})$ 比特。这改进了先前 $\widetilde{O}(n^{5/3})$ 的最佳界限，该界限是通过前述割查询算法的归约得到的。与此同时，已有研究观察到，通过连续优化技术可以实现 $\widetilde{O}(n^{3/2})$ 比特的随机协议；然而，这些方法与我们的组合方法有本质区别。