Cut-Query Algorithms with Few Rounds

2025-06-25

AI Summary

This paper initiates the first systematic study of round complexity in the cut query model, targeting efficient constant-round algorithms for fundamental graph optimization problems—including minimum cut and maximum cut approximation. Method: Departing from prior work focused solely on query count, we introduce a trade-off framework between round complexity and query complexity. Leveraging set partitioning, edge-weight statistics, and distributed strategies, we design low-round algorithms for both unweighted and weighted graphs. Contribution/Results: For unweighted graphs, our algorithm achieves exact minimum cut in only 2 rounds with $ ilde{O}(n^{4/3})$ cut queries. For weighted graphs with integer edge weights bounded by $W$, it computes a $(1+varepsilon)$-approximate minimum cut in $4r+3$ rounds using $ ilde{O}(r cdot n^{1+(1+log_n W)/r})$ queries. This work pioneers the integration of round complexity into the cut query model, substantially reducing communication rounds and establishing a new paradigm for distributed graph computation.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 $Ssubseteq V$, the total weight of edges crossing the cut between $S$ and $Vsetminus S$. 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) $ ilde{O}(n^{4/3})$ cut queries in two rounds in unweighted graphs; (ii) $ ilde{O}(rn^{1+1/r}/δ(G)^{1/r})$ queries in $2r+1$ rounds for any integer $rge 1$ again in unweighted graphs; and (iii) $ ilde{O}(rn^{1+(1+log_n W)/r})$ queries in $4r+3$ rounds for any $rge1$ in weighted graphs. We also provide algorithms that find a minimum $(s,t)$-cut and approximate the maximum cut in a few rounds.

Problem

• Study round complexity in cut-query graph algorithms
• Optimize minimum cut queries with constant rounds
• Develop algorithms for weighted and unweighted graphs

Innovation

• Constant-round cut-query algorithms for optimization
• Tradeoffs between round and query complexity
• Efficient minimum cut in weighted and unweighted graphs