# Cut-equivalent trees are optimal for min-cut queries

## Abstract

Min-Cut queries are fundamental: Preprocess an undirected edge-weighted graph, to quickly report a minimum-weight cut that separates a query pair of nodes s, t. The best data structure known for this problem simply builds a cut-equivalent tree, discovered 60 years ago by Gomory and Hu, who also showed how to construct it using n-1 minimum st-cut computations. Using state-of-the-art algorithms for minimum st-cut (Lee and Sidford, FOCS 2014), one can construct the tree in time tilde{O}(mn{3/2}), which is also the preprocessing time of the data structure. (Throughout, we focus on polynomially-bounded edge weights, noting that faster algorithms are known for small/u nit edge weights, and use n and m for the number of nodes and edges in the graph.) Our main result shows the following equivalence: Cut-equivalent trees can be constructed in near-linear time if and only if there is a data structure for Min-Cut queries with near-linear preprocessing time and polylogarithmic (amortized) query time, and even if the queries are restricted to a fixed source. That is, equivalent trees are an essentially optimal solution for Min-Cut queries. This equivalence holds even for every minor-closed family of graphs, such as bounded-treewidth graphs, for which a two-decade old data structure (Arikati, Chaudhuri, and Zaroliagis, J. Algorithms 1998) implies the first near-linear time construction of cut-equivalent trees. Moreover, unlike all previous techniques for constructing cut-equivalent trees, ours is robust to relying on approximation algorithms. In particular, using the almost-linear time algorithm for (1+ varepsilon)-approximate minimum st-cut (Kelner, Lee, Orecchia, and Sidford, SODA 2014), we can construct a (1+ varepsilon)-approximate flow-equivalent tree (which is a slightly weaker notion) in time n{2+o(1)}. This leads to the first (1+ varepsilon)-approximation for All-Pairs Max-Flow that runs in time n{2+o(1)}, and matches the output size almost-optimally.