Friendly Cut Sparsifiers and Faster Gomory-Hu Trees

TL;DR

This paper introduces new friendly cut sparsifiers that enable faster algorithms for constructing Gomory-Hu trees, significantly improving efficiency for dense graphs by leveraging novel sparsification techniques.

Contribution

The paper presents the concept of friendly cut sparsifiers and terminal sparsifiers, along with algorithms to compute them efficiently, leading to improved Gomory-Hu tree construction times.

Findings

Developed almost-linear time algorithms for friendly cut sparsifiers with  O(n AA tilde{O}(n \u0000AAA tilde{O}(n \u00A ext{sqrt}(k)) edges.

Enhanced Gomory-Hu tree algorithms to run in A tilde{O}(m + n^{1.75}) time for dense graphs.

Identified potential for further improvements under certain sparsification hypotheses.

Abstract

We devise new cut sparsifiers that are related to the classical sparsification of Nagamochi and Ibaraki [Algorithmica, 1992], which is an algorithm that, given an unweighted graph $G$ on $n$ nodes and a parameter $k$, computes a subgraph with $O(nk)$ edges that preserves all cuts of value up to $k$. We put forward the notion of a friendly cut sparsifier, which is a minor of $G$ that preserves all friendly cuts of value up to $k$, where a cut in $G$ is called friendly if every node has more edges connecting it to its own side of the cut than to the other side. We present an algorithm that, given a simple graph $G$, computes in almost-linear time a friendly cut sparsifier with $\tilde{O}(n \sqrt{k})$ edges. Using similar techniques, we also show how, given in addition a terminal set $T$, one can compute in almost-linear time a terminal sparsifier, which preserves the minimum $st$-cut between every pair of terminals, with $\tilde{O}(n \sqrt{k} + |T| k)$ edges. Plugging these sparsifiers into the recent $n^{2+o(1)}$-time algorithms for constructing a Gomory-Hu tree of simple graphs, along with a relatively simple procedure for handling the unfriendly minimum cuts, we improve the running time for moderately dense graphs (e.g., with $m=n^{1.75}$ edges). In particular, assuming a linear-time Max-Flow algorithm, the new state-of-the-art for Gomory-Hu tree is the minimum between our $(m+n^{1.75})^{1+o(1)}$ and the known $m n^{1/2+o(1)}$. We further investigate the limits of this approach and the possibility of better sparsification. Under the hypothesis that an $\tilde{O}(n)$-edge sparsifier that preserves all friendly minimum $st$-cuts can be computed efficiently, our upper bound improves to $\tilde{O}(m+n^{1.5})$ which is the best possible without breaking the cubic barrier for constructing Gomory-Hu trees in non-simple graphs.