A Simple Framework for Finding Balanced Sparse Cuts via APSP

TL;DR

This paper introduces a simple, intuitive algorithm for finding balanced sparse cuts in graphs using shortest-paths, combining a new multiplicative-weights framework with standard ball growing arguments, and achieves near-linear time complexity.

Contribution

It presents a novel, simple deterministic algorithm for balanced sparse cuts that matches state-of-the-art results with less complexity and easier analysis.

Findings

Runs in near-linear time $ ilde{O}(m^2/)$ for finding sparse cuts.

Achieves $ ilde{O}()$-sparse balanced cuts when a $$-sparse cut exists.

Simplifies previous complex algorithms while maintaining optimal theoretical guarantees.

Abstract

We present a very simple and intuitive algorithm to find balanced sparse cuts in a graph via shortest-paths. Our algorithm combines a new multiplicative-weights framework for solving unit-weight multi-commodity flows with standard ball growing arguments. Using Dijkstra's algorithm for computing the shortest paths afresh every time gives a very simple algorithm that runs in time $\widetilde{O}(m^2/\phi)$ and finds an $\widetilde{O}(\phi)$-sparse balanced cut, when the given graph has a $\phi$-sparse balanced cut. Combining our algorithm with known deterministic data-structures for answering approximate All Pairs Shortest Paths (APSP) queries under increasing edge weights (decremental setting), we obtain a simple deterministic algorithm that finds $m^{o(1)}\phi$-sparse balanced cuts in $m^{1+o(1)}/\phi$ time. Our deterministic almost-linear time algorithm matches the state-of-the-art in randomized and deterministic settings up to subpolynomial factors, while being significantly simpler to understand and analyze, especially compared to the only almost-linear time deterministic algorithm, a recent breakthrough by Chuzhoy-Gao-Li-Nanongkai-Peng-Saranurak (FOCS 2020).