Finding and Certifying Weighted Minimum Cuts with Cut Queries

Abstract

We study the deterministic and nondeterministic cut query complexity of minimum cut in undirected graphs with arbitrary nonnegative edge weights. For the nondeterministic case, we show that every $n$-vertex graph has an $O(n\log^2(2n))$-query certificate of its exact global minimum-cut value, such that any other nonnegatively weighted graph on the same vertex set that agrees on these queries has the same minimum cut value. A key ingredient is a way to certify a maximum $s$--$t$ flow value with only $O(n)$ cut queries. For the deterministic case we give an algorithm that finds an exact global minimum cut using $O(n^{3/2}\log^3(2n)(\log\log(2n))^2)$ cut queries. Previously no $O(n^{2-\varepsilon})$ deterministic bound was known for arbitrary nonnegatively weighted graphs. The algorithm follows Karger's two-respecting tree approach. The key new ingredient is a deterministic construction of a weak skeleton of the input graph $G$: an explicit weighted graph $H$ in which every minimum cut of $G$ is a $5/4$-approximate minimum cut.