Cut-Query Algorithms with Few Rounds
Yotam Kenneth-Mordoch, Robert Krauthgamer
TL;DR
This work analyzes the global minimum cut problem in the cut-query model with a focus on round complexity. It introduces three core techniques—edge sampling primitives, contraction-based graph reductions, and cut sparsification—to achieve multiple low-round algorithms with polylogarithmic query costs: a two-round unweighted minimum cut with $\tilde{O}(n^{4/3})$ queries, a $2r+1$-round unweighted variant with $\tilde{O}(r n^{1+1/r}/\delta(G)^{1/r})$ queries, and a $O(r)$-round weighted minimum cut with $\tilde{O}(r n^{1+(1+\log_n W)/r})$ queries. The approach also yields efficient algorithms for minimum $(s,t)$-cuts and MAX-CUT approximations, and has streaming implications via cut-sparsifier based reductions. The results demonstrate that polylogarithmic-round algorithms with polylogarithmic-per-round queries are achievable for fundamental graph problems in the cut-query model, providing a versatile toolkit (edge sampling, star- and out-contractions, forest packings, and monotone matrix methods) for further low-adaptivity design. These techniques pave the way for near-linear or better-than-baseline query complexity in a small number of rounds and connect cutting-edge submodular planning with practical streaming and dynamic-graph applications.
Abstract
In the cut-query model, the algorithm can access the input graph $G=(V,E)$ only via cut queries that report, given a set $S\subseteq V$, the total weight of edges crossing the cut between $S$ and $V\setminus S$. This model was introduced by Rubinstein, Schramm and Weinberg [ITCS'18] and its investigation has so far focused on the number of queries needed to solve optimization problems, such as global minimum cut. We turn attention to the round complexity of cut-query algorithms, and show that several classical problems can be solved in this model with only a constant number of rounds. Our main results are algorithms for finding a minimum cut in a graph, that offer different tradeoffs between round complexity and query complexity, where $n=|V|$ and $δ(G)$ denotes the minimum degree of $G$: (i) $\tilde{O}(n^{4/3})$ cut queries in two rounds in unweighted graphs; (ii) $\tilde{O}(rn^{1+1/r}/δ(G)^{1/r})$ queries in $2r+1$ rounds for any integer $r\ge 1$ again in unweighted graphs; and (iii) $\tilde{O}(rn^{1+(1+\log_n W)/r})$ queries in $4r+3$ rounds for any $r\ge1$ in weighted graphs. We also provide algorithms that find a minimum $(s,t)$-cut and approximate the maximum cut in a few rounds.