Subcubic Algorithms for Gomory-Hu Tree in Unweighted Graphs

TL;DR

This paper introduces the first subcubic algorithm for constructing Gomory-Hu trees in unweighted graphs, significantly improving the efficiency of all-pairs max-flow computations.

Contribution

It presents the first subcubic-time algorithm for Gomory-Hu tree construction in simple graphs, reducing the complexity from cubic to approximately $O(n^{2.5})$.

Findings

Achieves $ ilde{O}(n^{2.5})$ time complexity for Gomory-Hu tree construction.

Breaks the cubic-time barrier for All-Pairs Max-Flow in simple graphs.

Reduces the problem to $ ilde{O}( oot{n} ext{)}$ Max-Flow computations.

Abstract

Every undirected graph $G$ has a (weighted) cut-equivalent tree $T$, commonly named after Gomory and Hu who discovered it in 1961. Both $T$ and $G$ have the same node set, and for every node pair $s,t$, the minimum $(s,t)$-cut in $T$ is also an exact minimum $(s,t)$-cut in $G$. We give the first subcubic-time algorithm that constructs such a tree for a simple graph $G$ (unweighted with no parallel edges). Its time complexity is $\tilde{O}(n^{2.5})$, for $n=|V(G)|$; previously, only $\tilde{O}(n^3)$ was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using $n-1$ queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to $\tilde{O}(\sqrt{n})$ Max-Flow computations on $n$-node graphs.