Computing exact minimum cuts without knowing the graph

TL;DR

This paper introduces query-efficient algorithms for exactly computing minimum cuts in unweighted, undirected graphs without prior graph knowledge, significantly reducing the number of queries needed compared to learning the entire graph.

Contribution

It presents novel algorithms that find exact minimum s-t cuts and global minimum cuts with fewer queries than full graph reconstruction, inspired by submodular function minimization.

Findings

Exact s-t cut computed with ~O(n^{5/3}) queries

Global min cut computed with ~O(n) queries

Graph learning requires ~Θ(n^2) queries

Abstract

We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query $S \subset V$, the oracle returns the size of the cut between $S$ and $V \setminus S$. We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with $\tilde{O}(n^{5/3})$ queries, and computing an exact global minimum cut of $G$ with only $\tilde{O}(n)$ queries (while learning the graph requires $\tilde{\Theta}(n^2)$ queries).