A Simple and Fast Algorithm for Fair Cuts
Jason Li, Owen Li
TL;DR
The paper addresses computing fair cuts, a robust generalization of approximate min-cut, by presenting a randomized algorithm that runs in $\tilde{O}(m/\epsilon)$ time to produce a $(1+O(\epsilon \log n))$-fair $(s,t)$-cut on undirected graphs with integral polynomial capacities. The approach iteratively applies Sherman's approximate max-flow/min-cut algorithm on directed residual graphs, guaranteeing progress by reducing the relevant residual cut capacity by a constant factor in each iteration, and augmenting flows to certify near-fair cuts. The key contribution is bridging the gap between fair-cut and standard min-cut runtimes, enabling near-linear-time fair-cut procedures and enabling parallelizable fair-cut-based primitives for Steiner min-cut, Gomory-Hu trees, and expander decompositions. This yields a practically impactful improvement, matching the speed of standard $(1+\epsilon)$-approximate min-cut and enhancing robustness without sacrificing efficiency or parallelizability.
Abstract
We present a simple and faster algorithm for computing fair cuts on undirected graphs, a concept introduced in recent work of Li et al. (SODA 2023). Informally, for any parameter $ε>0$, a $(1+ε)$-fair $(s,t)$-cut is an $(s,t)$-cut such that there exists an $(s,t)$-flow that uses $1/(1+ε)$ fraction of the capacity of every edge in the cut. Our algorithm computes a $(1+ε)$-fair cut in $\tilde O(m/ε)$ time, improving on the $\tilde O(m/ε^3)$ time algorithm of Li et al. and matching the $\tilde O(m/ε)$ time algorithm of Sherman (STOC 2017) for standard $(1+ε)$-approximate min-cut. Our main idea is to run Sherman's approximate max-flow/min-cut algorithm iteratively on a (directed) residual graph. While Sherman's algorithm is originally stated for undirected graphs, we show that it provides guarantees for directed graphs that are good enough for our purposes.