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Breaking the n-Pass Barrier: A Streaming Algorithm for Maximum Weight Bipartite Matching [article]

S. Cliff Liu, Zhao Song, Hengjie Zhang
2021 arXiv   pre-print
To the best of our knowledge, this is the first work that implements the SDD solvers and IPMs in the streaming model in O(n) spaces for graph matrices; previous IPM algorithms only focus on optimizing  ...  Given a weighted bipartite graph with n vertices and m edges, the maximum weight bipartite matching problem is to find a set of vertex-disjoint edges with the maximum weight.  ...  For moderately dense graph, a better algorithm is given in [BLN + 20] very recently, which runs in O(m + n 1.5 ) time and takes Ω(m) space.  ... 
arXiv:2009.06106v2 fatcat:bkjmromopvg6zfubstu444dg2e

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions [article]

Timothy Chu, Yu Gao, Richard Peng, Sushant Sachdeva, Saurabh Sawlani, Junxing Wang
2018 arXiv   pre-print
graph sparsification, we show the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs.  ...  A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 n, and at most 2n extra edges.  ...  Acknowledgments We thank John Peebles for many insightful discussions, David Durfee for pointing out several key details in the interaction with determinant estimation from Appendix B, and Di Wang for  ... 
arXiv:1805.12051v1 fatcat:6jz5yc74srfydjhrcept4ohwnm

Nearly-Linear Work Parallel SDD Solvers, Low-Diameter Decomposition, and Low-Stretch Subgraphs

Guy E. Blelloch, Anupam Gupta, Ioannis Koutis, Gary L. Miller, Richard Peng, Kanat Tangwongsan
2013 Theory of Computing Systems  
We apply this subgraph construction to derive a parallel linear solver.  ...  To this end, we first develop a parallel decomposition algorithm that in O(m log O(1) n) work and polylogarithmic depth, partitions a graph with n nodes and m edges into components with polylogarithmic  ...  via interior point methods described in [4, 24, 32] , where U is the ratio of the largest edge weight to the smallest nonzero edge weight in the graph.  ... 
doi:10.1007/s00224-013-9444-5 fatcat:crkr2culg5aytl7hh66fd52n34

Correlation Clustering in Data Streams [article]

Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, Anthony Wirth
2018 arXiv   pre-print
The stream consists of updates to the edge weights of a graph on n nodes and the goal is to find a node-partition such that the end-points of negative-weight edges are typically in different clusters whereas  ...  Unfortunately, the standard LP and SDP formulations are not obviously solvable in O(n·polylog n)-space.  ...  They are based on linear sketches and incorporate ideas from work on constructing graph sparsifiers via linear sketches.  ... 
arXiv:1812.02023v1 fatcat:5rn76brcofh7zjlas7rcjwcesq

Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification

Deeksha Adil, Brian Bullins, Rasmus Kyng, Sushant Sachdeva, Nikhil Bansal, Emanuela Merelli, James Worrell
2021
find (1+2^{-poly(log n)}) approximations for weighted 𝓁_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = ω(1), which is almost-linear for graphs that are slightly dense  ...  We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (𝓁²₂ + 𝓁^p_p) flow or voltage objectives on graphs.  ...  by O(n log n). ◀ -Time ℓ p -Norm Solvers via Sparsification 0 2 1 9:8 Almost-Linear-Time ℓ p -Norm Solvers via Sparsification 2.  ... 
doi:10.4230/lipics.icalp.2021.9 fatcat:wxtvuhv6t5chfcvbdwqumiymy4

Electrical Flows over Spanning Trees [article]

Swati Gupta, Ali Khodabakhsh, Hassan Mortagy, Evdokia Nikolova
2020 arXiv   pre-print
To obtain the result for general graphs, we propose a new method for (approximate) spectral graph sparsification, which may be of independent interest.  ...  The tree requirement on the support of the flow is motivated by operational constraints in electricity distribution networks.  ...  grid embedding of planar graphs, and the anonymous referees for numerous useful suggestions to improve this manuscript.  ... 
arXiv:1909.04759v3 fatcat:hlzwwbnhwjf6ppmjpsuiedx6l4

Unifying Width-Reduced Methods for Quasi-Self-Concordant Optimization [article]

Deeksha Adil, Brian Bullins, Sushant Sachdeva
2021 arXiv   pre-print
et al., SODA'19], in terms of improving the iteration complexity from O(m^1/2) to Õ(m^1/3), where m is the number of rows of the design matrix, and where each iteration amounts to a linear system solve  ...  In order to do so, we develop a unified width reduction method for carefully handling these losses based on a more general set of potentials.  ...  Almost-lineartime weighted ℓ p -norm solvers in slightly dense graphs via sparsification, 2021. Brian Bullins. Highly smooth minimization of non-smooth problems.  ... 
arXiv:2107.02432v1 fatcat:e57ppynnyrghtgr5fr3psgqsam

Accelerated multigrid for graph Laplacian operators

Pietro Dell'Acqua, Antonio Frangioni, Stefano Serra-Capizzano
2015 Applied Mathematics and Computation  
In this setting, a linear system involving the weighted Laplacian of the underlying network has to be solved at each iteration, with varying vector of arc weights Θ (cf. relation (2) ).  ...  We consider multigrid type techniques for the numerical solution of large linear systems, whose coefficient matrices show the structure of (weighted) graph Laplacian operators.  ...  Finally, we acknowledge that the work of the third author has been partly supported via Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation, in collaboration with the Royal Swedish Academy  ... 
doi:10.1016/j.amc.2015.08.033 fatcat:a7z6buplnnhlfcw4xtrwwcr2wq

Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers [article]

Jan van den Brand, Yu Gao, Arun Jambulapati, Yin Tat Lee, Yang P. Liu, Richard Peng, Aaron Sidford
2021 arXiv   pre-print
More efficient dynamic spectral vertex sparsification, achieved by faster length estimation of random walks in weighted graphs using Morris counters [Morris 1978, Nelson-Yu 2020]. 2.  ...  We make several advances broadly related to the maintenance of electrical flows in weighted graphs undergoing dynamic resistance updates, including: 1.  ...  Flows in almost linear time via adaptive preconditioning.  ... 
arXiv:2112.00722v1 fatcat:v5fv24yzpbdrzlkeso4ofz5haq

Lecture Notes on Spectral Graph Methods [article]

Michael W. Mahoney
2016 arXiv   pre-print
These are lecture notes that are based on the lectures from a class I taught on the topic of Spectral Graph Methods at UC Berkeley during the Spring 2015 semester.  ...  Computations, e.g., nearly linear time Laplacian solvers and graph algorithms in the language of linear algebra.  ...  They used this to find good well-balanced graph partitions in nearly-linear time, which they then used as a subroutine in their efforts to develop nearly linear time solvers for Laplacian-based linear  ... 
arXiv:1608.04845v1 fatcat:ppy6mlmfsvfcxedriwnndv6ztq

Speeding Up Sparsification using Inner Product Search Data Structures [article]

Zhao Song, Zhaozhuo Xu, Lichen Zhang
2022 arXiv   pre-print
We obtain improved running time for the following problems. * For constructing linear-sized spectral sparsifier (Batson, Spielman and Srivastava, 2012), all the existing deterministic algorithms require  ...  In this work, we provide the first deterministic algorithm that breaks that barrier which runs in O(d^ω+1) time, where ω is the exponent of matrix multiplication. * For one-sided Kadison-Singer-typed discrepancy  ...  Linear-Sized Spectral Sparsification via Positive Inner Product Search In this section, we study the linear-sized spectral sparsification problem for a V ∈ R m×d matrix. • In Section 9.1, we setup the  ... 
arXiv:2204.03209v1 fatcat:jgxikkra7jbp7fgk744scli2fa

Faster Parallel Algorithm for Approximate Shortest Path [article]

Jason Li
2022 arXiv   pre-print
We present the first m polylog(n) work, polylog(n) time algorithm in the PRAM model that computes (1+ϵ)-approximate single-source shortest paths on weighted, undirected graphs.  ...  To improve readability, the paper is almost entirely self-contained, save for several staple theorems in algorithms and combinatorics.  ...  The author thanks Alexandr Andoni, Cliff Stein, and Peilin Zhong for pointing out a nontrivial omission of the first version of this paper, namely an issue in rounding the transshipment flow to an "expected  ... 
arXiv:1911.01626v6 fatcat:itluz3bbardh7babmcz4qqad3a

Semi-Streaming Bipartite Matching in Fewer Passes and Optimal Space [article]

Sepehr Assadi, Arun Jambulapati, Yujia Jin, Aaron Sidford, Kevin Tian
2021 arXiv   pre-print
We provide O(ϵ^-1)-pass semi-streaming algorithms for computing (1-ϵ)-approximate maximum cardinality matchings in bipartite graphs.  ...  Further, we leverage these techniques to obtain improvements for streaming variants of approximate linear programming, optimal transport, exact matching, transshipment, and shortest path problems.  ...  time linear in the sparsity of the matrix A).  ... 
arXiv:2011.03495v4 fatcat:tvmrf5dn3nhcfhu7kpnqppzwzu

An Improved Cutting Plane Method for Convex Optimization, Convex-Concave Games and its Applications [article]

Haotian Jiang, Yin Tat Lee, Zhao Song, Sam Chiu-wai Wong
2020 arXiv   pre-print
log (κ) + n^ω+1log (κ)) time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on n, where ω < 2.373 is the exponent of matrix multiplication and SO is the time for oracle evaluation. ∙  ...  This improves upon Lee-Sidford-Wong's O( SO· n log (κ) + n^3 log^O(1) (κ)) time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on κ.  ...  This project was supported in part by NSF awards CCF-1749609, CCF-1740551, DMS-1839116, and Microsoft Research Faculty Fellowship.  ... 
arXiv:2004.04250v1 fatcat:imxhan7abfbbhbhygdkd6nbhsu

Quantum Algorithms for Unsupervised Machine Learning and Neural Networks [article]

Jonas Landman
2021 arXiv   pre-print
This allows us to define many fundamental procedures, in particular in vector and graph analysis. We will also present new quantum algorithms for neural networks, or deep learning.  ...  In this thesis, we investigate whether quantum algorithms can be used in the field of machine learning for both long and near term quantum computers.  ...  the authors performed tasks like sparsification of the graph faster than classical algorithms.  ... 
arXiv:2111.03598v1 fatcat:k7b5oct53zcrdewj4xybvem3je
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