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

2025-10-20

AI Summary

This paper studies the minimum $s$-$t$ cut problem on unweighted, undirected graphs in the cut query model, where algorithms may only access the graph via an oracle that returns the cut size $|delta(S)|$ for any queried vertex subset $S subseteq V$, without direct access to edge structure. We present the first randomized algorithm with $widetilde{O}(n^{8/5})$ cut queries, improving significantly over the prior best bound of $widetilde{O}(n^2)$. Our method introduces a query-efficient recursive greedy subroutine that incrementally uncovers graph structure and simultaneously augments flow—without explicitly constructing augmenting paths. The approach is fully combinatorial, avoiding linear programming or numerical optimization. As a byproduct, we obtain a deterministic two-party communication protocol for minimum $s$-$t$ cut with $widetilde{O}(n^{11/7})$ communication complexity, providing a new tool at the intersection of cut queries and distributed computation.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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 $widetildeΘ(n^2)$ 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.

Problem

• Computing minimum s-t cuts using fewer cut queries in graphs
• Improving cut-query complexity for minimum s-t cut algorithms
• Developing efficient deterministic communication protocols for cut problems

Innovation

• Randomized algorithm reduces cut queries complexity
• Incremental edge revelation increases flow efficiently
• Deterministic combinatorial protocol improves communication bound