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Vertex Sparsifiers for c-Edge Connectivity
[article]

2019
*
arXiv
*
pre-print

We show the existence of O(f(

arXiv:1910.10359v1
fatcat:s5zje623vffozkbelss6qjgqn4
*c*)k) sized*vertex**sparsifiers*that preserve all*edge*-*connectivity*values up to*c*between a set of k terminal vertices, where f(*c*) is a function that only depends on*c*, the ... It implies that*for*constant values of*c*, an offline sequence of*edge*insertions/deletions and*c*-*edge*-*connectivity*queries can be answered in polylog time per operation. ... Acknowledgements We thank Gramoz Goranci, Jakub Lacki, Thatchaphol Saranurak, and Xiaorui Sun*for*multiple enlightening discussions on this topic. ...##
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Improved Guarantees for Vertex Sparsification in Planar Graphs
[article]

2017
*
arXiv
*
pre-print

In this work we study

arXiv:1702.01136v1
fatcat:pghxyimrvfgptl7vgq5ubynyai
*Vertex**Sparsifiers*, i.e.,*sparsifiers*whose goal is to reduce the number of vertices. ... It improves the previous best-known bound of O(k^22^2k)*for*cut and flow*sparsifiers*by an exponential factor, and matches an Ω(k^2) lower-bound*for*this class of graphs. ... Delta-Wye transformation: Delete the*edges*of a triangle*connecting*x, y and z, introduce a new non-terminal*vertex*w and add new*edges*(w, x), (w, y) and (w, z) with*edge*capacities*c*(x, y) +*c*(x, z), ...##
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Vertex Sparsification in Trees
[article]

2016
*
arXiv
*
pre-print

Given an unweighted tree T=(V,E) with terminals K ⊂ V, we show how to obtain a 2-quality

arXiv:1612.03017v1
fatcat:eggbm2pinranldvaqdabdakm2i
*vertex*flow and cut*sparsifier*H with V_H = K. ... First, we show how to obtain a 2-quality flow*sparsifier*with V_H = K*for*such graphs. ... Then the graph H ′ = i α i · H i is a*vertex*flow*sparsifier**for*G. ...##
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Levelwise Mesh Sparsification for Shortest Path Queries
[chapter]

2010
*
Lecture Notes in Computer Science
*

We consider regions of several sizes, and construct the

doi:10.1007/978-3-642-17517-6_13
fatcat:pgp7qb2f2fczvhq2iilusmdpcy
*sparsified*network*for*each region composed of*edges*which are parts of shortest paths of vertices far from the region. ...*For*each query, the sparse network is constructed by combining the*sparsified*networks*for*which the origin and the destination are distant. ... Moreover, by considering regions with different sizes,*for*example the rectangles whose*edge*is of length*c*· 2 k*for*some*c*, we can use larger region with more sparse network*for*distant vertices. ...##
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Grid Induced Minor Theorem for Graphs of Small Degree
[article]

2022
*
arXiv
*
pre-print

A graph H is an induced minor of a graph G if H can be obtained from G by

arXiv:2203.13233v1
fatcat:hkhwicm6ybfh7exv6izewzcube
*vertex*deletions and*edge*contractions. ... It also implies that*for*any fixed planar graph H, there is a subexponential time algorithm*for*maximum weight independent set on H-induced-minor-free graphs. ... We call a graph*sparsifiable*if every*vertex*of it is*sparsifiable*. ...##
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On vertex sparsifiers with Steiner nodes

2012
*
Proceedings of the 44th symposium on Theory of Computing - STOC '12
*

Given an undirected graph G = (V, E) with

doi:10.1145/2213977.2214039
dblp:conf/stoc/Chuzhoy12
fatcat:qfef6d56xbftzcllw4pgitxzvm
*edge*capacities*c*e ≥ 1*for*e ∈ E and a subset T of k vertices called terminals, we say that a graph H is a quality-q cut*sparsifier**for*G iff T ⊆ V (H), and*for*...*For*this setting, efficient algorithms are known*for*constructing quality-O(log k/ log log k) cut and flow*vertex**sparsifiers*, as well as a lower bound ofΩ( √ log k) on the quality of any flow or cut*sparsifier*... Acknowledgements The author thanks Yury Makarychev and Konstantin Makarychev*for*many interesting discussions about*vertex**sparsifiers*. ...##
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On Vertex Sparsifiers with Steiner Nodes
[article]

2012
*
arXiv
*
pre-print

*For*this setting, efficient algorithms are known

*for*constructing quality-O( k/ k) cut and flow

*vertex*

*sparsifiers*, as well as a lower bound of Ω̃(√( k)) on the quality of any flow or cut

*sparsifier*. ... Given an undirected graph G=(V,E) with

*edge*capacities c_e≥ 1

*for*e∈ E and a subset T of k vertices called terminals, we say that a graph H is a quality-q cut

*sparsifier*

*for*G iff T⊆ V(H), and

*for*any ... Acknowledgements The author thanks Yury Makarychev and Konstantin Makarychev

*for*many interesting discussions about

*vertex*

*sparsifiers*. ...

##
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On Fully Dynamic Graph Sparsifiers

2016
*
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
*

We initiate the study of dynamic algorithms

doi:10.1109/focs.2016.44
dblp:conf/focs/AbrahamDKKP16
fatcat:tu35ze66lncbdbg32fjv63az5u
*for*graph sparsification problems and obtain fully dynamic algorithms, allowing both*edge*insertions and*edge*deletions, that take polylogarithmic time after ... Second, we give a fully dynamic algorithm*for*maintaining a (1 ±ϵ) -cut*sparsifier*with worst-case update time poly(n, ϵ^-1). Both*sparsifiers*have size n · poly(n, ϵ^-1). ... Proof of Theorem 8.10 : Any*edge*insertion/deletion inG requires O(poly(log n, −1 )) update time*for*G and VC from Lemma 8.13. ...##
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Degree-3 Treewidth Sparsifiers
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

This is closely related to the open question of computing small good-quality

doi:10.1137/1.9781611973730.19
dblp:conf/soda/ChekuriC15
fatcat:mqjnteicoffrvgildnjatz7w6u
*vertex*-cut*sparsifiers*that are also minors of the original graph. ... Informally, given a graph G of treewidth k, a treewidth*sparsifier*H is a minor of G, whose treewidth is close to k, |V (H)| is small, and the maximum*vertex*degree in H is bounded. ... Acknowledgement: We thank Paul Seymour*for*posing the question of the existence of degree-3 treewidth*sparsifiers*to us. ...##
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Degree-3 Treewidth Sparsifiers
[article]

2014
*
arXiv
*
pre-print

This is closely related to the open question of computing small good-quality

arXiv:1410.1016v1
fatcat:rc43n4je7ffg7c4sbfipbalpoq
*vertex*-cut*sparsifiers*that are also minors of the original graph. ... Informally, given a graph G of treewidth k, a treewidth*sparsifier*H is a minor of G, whose treewidth is close to k, |V(H)| is small, and the maximum*vertex*degree in H is bounded. ... Acknowledgement: We thank Paul Seymour*for*posing the question of the existence of degree-3 treewidth*sparsifiers*to us. ...##
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Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers
[article]

2020
*
arXiv
*
pre-print

Result (3) is obtained by invoking the random-walk based spectral

arXiv:2005.02368v1
fatcat:rnksl5fksjfrxixq4n2pdxutcy
*vertex**sparsifier*by [Durfee et al. ... In particular, we develop a technique that, given any problem that admits a certain notion of*vertex**sparsifiers*, gives data structures that maintain approximate solutions in sub-linear update and query ... Concurrent to our result there have also been several recent developments on utilizing*vertex**sparsifiers*to maintain*c*-*edge**connectivity**for*small values of*c*[PSS19, CDLV19, LPS19, JS20]. ...##
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Vertex Sparsification for Edge Connectivity in Polynomial Time
[article]

2021
*
arXiv
*
pre-print

(SODA 2021) introduced a relaxation called

arXiv:2011.15101v2
fatcat:na2csogaq5bx7bzzxgjqxempta
*connectivity*-*c*mimicking networks, which asks to construct a*vertex**sparsifier*which preserves*connectivity*among k terminals exactly up to the value of*c*, and ... (SODA 2021)*for*any*c*≥log n, whose runtimes depended exponentially on*c*. ... Acknowledgments The author would like to thank Yunbum Kook*for*feedback on an earlier version of this manuscript, and Richard Peng*for*useful discussions and encouragement. ...##
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Vertex and Hyperedge Connectivity in Dynamic Graph Streams

2015
*
Proceedings of the 34th ACM Symposium on Principles of Database Systems - PODS '15
*

We present the first linear sketches

doi:10.1145/2745754.2745763
dblp:conf/pods/GuhaMT15
fatcat:o2yg5pjzdrfghdnjbghvtlhey4
*for*estimating*vertex**connectivity*and constructing hypergraph*sparsifiers*. ...*Vertex**connectivity*exhibits markedly different combinatorial structure than*edge**connectivity*and appears to be harder to estimate in the dynamic graph stream model. ... We thank Jennifer Chayes*for*prompting us to investigate hypergraph*connectivity*. References ...##
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On Mimicking Networks Representing Minimum Terminal Cuts
[article]

2012
*
arXiv
*
pre-print

Specifically, the

arXiv:1207.6371v1
fatcat:jmkj7tocuvhs5jbbl6qs4g25aa
*vertex*set of the*sparsifier*V_H contains the set of terminals K and*for*every bipartition U, K-U of the terminals K, the size of the minimum cut separating U from K-U in G is exactly ... More precisely, the best known lower bound is k+1*for*graphs with k terminals (Chaudhuri et al. 2000). ... A*vertex**sparsifier*H*for*graph G and terminal set K is a mimicking network if Q*C*(H) = 1. Nearly all existing constructions of*vertex**sparsifiers*are based on*edge*-contractions. ...##
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Single pass sparsification in the streaming model with edge deletions
[article]

2012
*
arXiv
*
pre-print

[SODA'12] to estimate

arXiv:1203.4900v1
fatcat:qjjlfeaoonhsjgafpjnhw6aebq
*edge**connectivity*together with a novel application of sampling with limited independence and sparse recovery to produce the*edges*of the*sparsifier*. ... Previous constructions either required multiple passes or were unable to handle*edge*deletions. We use Õ(1/^2) time*for*each stream update and Õ(n/^2) time to construct a*sparsifier*. ... Recall that*for*all u ∈ V (G) one has E[|E ′ ∩ E u |] ≤ 4γ log n. Fix a cut (*C*, V \*C*).*For*each*vertex*u ∈*C*let X u = (u,v)∈Eu,v ∈*C*X u,v . ...
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