Cut-Query Algorithms with Few Rounds

TL;DR

This paper introduces algorithms for the cut-query model that solve minimum cut problems with a small number of rounds, balancing query complexity and round complexity in both weighted and unweighted graphs.

Contribution

It presents novel algorithms for minimum cut in the cut-query model with constant or few rounds, improving efficiency over previous methods.

Findings

Constant round algorithms for minimum cut in unweighted graphs

Tradeoff algorithms between query and round complexity

Algorithms for minimum (s,t)-cut and approximate max cut

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 $\delta(G)$ denotes the minimum degree of $G$: (i) $\tilde{O}(n^{4/3})$ cut queries in two rounds in unweighted graphs; (ii) $\tilde{O}(rn^{1+1/r}/\delta(G)^{1/r})$ 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.