Faster Cut Sparsification of Weighted Graphs
Sebastian Forster, Tijn de Vos

TL;DR
This paper introduces a faster algorithm for computing cut sparsifiers of weighted graphs, improving efficiency and extending results to graphs with unbounded weights, leading to faster approximate min-cut algorithms.
Contribution
It presents a new algorithm for cut sparsification that achieves improved running time and handles unbounded weights, advancing the state of the art in graph sparsification.
Findings
Achieves $O(m imes ext{min}( ext{alpha}(n) ext{log}(m/n), ext{log}(n)))$ time for sparsification.
Provides the best known results for cut sparsification with unbounded weights.
Enables the fastest approximate min-cut algorithms for weighted graphs.
Abstract
A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of . This paper considers computing cut sparsifiers of weighted graphs of size . Our algorithm computes such a sparsifier in time , both for graphs with polynomially bounded and unbounded integer weights, where is the functional inverse of Ackermann's function. This improves upon the state of the art by Bencz\'ur and Karger (SICOMP 2015), which takes time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate…
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