All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

TL;DR

This paper presents a near-linear time algorithm for constructing Gomory-Hu trees in surface-embedded graphs, enabling fast minimum cut queries with improved efficiency over previous methods.

Contribution

It introduces the first sub-quadratic algorithm for all-pairs minimum cuts in bounded genus graphs, extending planar graph techniques to more complex surface-embedded graphs.

Findings

Preprocessing time is $O(n ext{log}^3 n)$ for bounded genus graphs.

Constructs a Gomory-Hu tree with $O(n)$ space.

Answers minimum cut queries in constant time.

Abstract

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$.