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

TL;DR

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 is presented.

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.

Figures (4)

• Figure 1: Shown on the left is a graph $G$ with a flow graph $F$ of size 2 indicated by the union of red and blue edges. Shown on the right is $G_F$ which is the residual graph of $G$ induced by $F$.
• Figure 2: Shown above is a graph $H_F$ with its partial labels given a fixed $C$. The shaded vertices ($s$ and $t$) are already given a label. The red and blue edges represent $E_2$ which is the reversal of a flow graph $F$ of size 2. The black edges represent undirected unit-capacitated edges, each of which viewed as two directed edges of weight 1. The wavy edges represent the fixing of $C$. Step 1 uniquely labels vertices in $V(C)$ according to $C$. Such unique label is indicated by the label within each node itself.
• Figure 3: Continuing from Step 1, the shaded vertices are already given a fixed label. Shown above one of the 4 possible consistent labeling to a segment of length $l = 4$ on the blue flow. The remaining 3 possibilities are as follows: 's' $\leftarrow$ 's' $\leftarrow$ 's'; 's' $\leftarrow$ 't' $\leftarrow$ 't'; 't' $\leftarrow$ 't' $\leftarrow$ 't'.
• Figure 4: Continuing from Step 2, the shaded vertices are already given a fixed label. Shown above is the only possible label to the remaining (unshaded) vertices

Theorems & Definitions (95)

• Theorem 1.0: Main result #1
• Theorem 1.0: Main result #2
• Lemma 2.1: Lemma 5.4 of RubinsteinSW18
• Lemma 2.1
• Proposition 2.2
• Theorem 2.3: Max-Flow Min-Cut Theorem
• Lemma 2.4
• Theorem 2.4
• Lemma 2.5: Theorem 3.5 of RubinsteinSW18
• Corollary 2.5