Faster Cut Sparsification of Weighted Graphs

by   Sebastian Forster, et al.

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of (1±ϵ). This paper considers computing cut sparsifiers of weighted graphs of size O(nlog (n)/ϵ^2). Our algorithm computes such a sparsifier in time O(m·min(α(n)log(m/n),log (n))), where α(·) is the functional inverse of Ackermann's function, both for graphs with polynomially bounded and unbounded integer weights. This improves upon the state of the art by Benczúr and Karger (SICOMP 2015), which takes O(mlog^2 (n)) rounds. 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 min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. (SICOMP 2019) for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi-Ibaraki (NI) forest packing. MSF packings have previously been used by Abraham at al.(FOCS 2016) in the dynamic setting, and are defined as follows: an M-partial MSF packing of G is a set ℱ={F_1, …, F_M}, where F_i is a maximum spanning forest in G∖⋃_j=1^i-1F_j. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.



There are no comments yet.


page 1

page 2

page 3

page 4


An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs

In this paper, we consider the task of computing an independent set of m...

Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms

Consider the following 2-respecting min-cut problem. Given a weighted g...

Finding cuts of bounded degree: complexity, FPT and exact algorithms, and kernelization

A matching cut is a partition of the vertex set of a graph into two sets...

Cut-Equivalent Trees are Optimal for Min-Cut Queries

Min-Cut queries are fundamental: Preprocess an undirected edge-weighted ...

Approximate Gomory-Hu Tree Is Faster Than n-1 Max-Flows

The Gomory-Hu tree or cut tree (Gomory and Hu, 1961) is a classic data s...

Distributed Weighted Min-Cut in Nearly-Optimal Time

Minimum-weight cut (min-cut) is a basic measure of a network's connectiv...

Connected Components on a PRAM in Log Diameter Time

We present an O(log d + loglog_m/n n)-time randomized PRAM algorithm for...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.