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Vertex Sparsifiers for c-Edge Connectivity [article]

Yang P. Liu and Richard Peng and Mark Sellke
2019 arXiv   pre-print
We show the existence of O(f(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.  ... 
arXiv:1910.10359v1 fatcat:s5zje623vffozkbelss6qjgqn4

Improved Guarantees for Vertex Sparsification in Planar Graphs [article]

Gramoz Goranci, Monika Henzinger, Pan Peng
2017 arXiv   pre-print
In this work we study 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),  ... 
arXiv:1702.01136v1 fatcat:pghxyimrvfgptl7vgq5ubynyai

Vertex Sparsification in Trees [article]

Gramoz Goranci, Harald Raecke
2016 arXiv   pre-print
Given an unweighted tree T=(V,E) with terminals K ⊂ V, we show how to obtain a 2-quality 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.  ... 
arXiv:1612.03017v1 fatcat:eggbm2pinranldvaqdabdakm2i

Levelwise Mesh Sparsification for Shortest Path Queries [chapter]

Yuichiro Miyamoto, Takeaki Uno, Mikio Kubo
2010 Lecture Notes in Computer Science  
We consider regions of several sizes, and construct the 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.  ... 
doi:10.1007/978-3-642-17517-6_13 fatcat:pgp7qb2f2fczvhq2iilusmdpcy

Grid Induced Minor Theorem for Graphs of Small Degree [article]

Tuukka Korhonen
2022 arXiv   pre-print
A graph H is an induced minor of a graph G if H can be obtained from G by 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.  ... 
arXiv:2203.13233v1 fatcat:hkhwicm6ybfh7exv6izewzcube

On vertex sparsifiers with Steiner nodes

Julia Chuzhoy
2012 Proceedings of the 44th symposium on Theory of Computing - STOC '12  
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  ...  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.  ... 
doi:10.1145/2213977.2214039 dblp:conf/stoc/Chuzhoy12 fatcat:qfef6d56xbftzcllw4pgitxzvm

On Vertex Sparsifiers with Steiner Nodes [article]

Julia Chuzhoy
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.  ... 
arXiv:1204.2844v1 fatcat:eidfp4q6gfdy5lmjchhcsw5aeu

On Fully Dynamic Graph Sparsifiers

Ittai Abraham, David Durfee, Ioannis Koutis, Sebastian Krinninger, Richard Peng
2016 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)  
We initiate the study of dynamic algorithms 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.  ... 
doi:10.1109/focs.2016.44 dblp:conf/focs/AbrahamDKKP16 fatcat:tu35ze66lncbdbg32fjv63az5u

Degree-3 Treewidth Sparsifiers [chapter]

Chandra Chekuri, Julia Chuzhoy
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 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.  ... 
doi:10.1137/1.9781611973730.19 dblp:conf/soda/ChekuriC15 fatcat:mqjnteicoffrvgildnjatz7w6u

Degree-3 Treewidth Sparsifiers [article]

Chandra Chekuri, Julia Chuzhoy
2014 arXiv   pre-print
This is closely related to the open question of computing small good-quality 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.  ... 
arXiv:1410.1016v1 fatcat:rc43n4je7ffg7c4sbfipbalpoq

Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers [article]

Li Chen, Gramoz Goranci, Monika Henzinger, Richard Peng, Thatchaphol Saranurak
2020 arXiv   pre-print
Result (3) is obtained by invoking the random-walk based spectral 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].  ... 
arXiv:2005.02368v1 fatcat:rnksl5fksjfrxixq4n2pdxutcy

Vertex Sparsification for Edge Connectivity in Polynomial Time [article]

Yang P. Liu
2021 arXiv   pre-print
(SODA 2021) introduced a relaxation called 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.  ... 
arXiv:2011.15101v2 fatcat:na2csogaq5bx7bzzxgjqxempta

Vertex and Hyperedge Connectivity in Dynamic Graph Streams

Sudipto Guha, Andrew McGregor, David Tench
2015 Proceedings of the 34th ACM Symposium on Principles of Database Systems - PODS '15  
We present the first linear sketches 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  ... 
doi:10.1145/2745754.2745763 dblp:conf/pods/GuhaMT15 fatcat:o2yg5pjzdrfghdnjbghvtlhey4

On Mimicking Networks Representing Minimum Terminal Cuts [article]

Arindam Khan, Prasad Raghavendra, Prasad Tetali, László A. Végh
2012 arXiv   pre-print
Specifically, the 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.  ... 
arXiv:1207.6371v1 fatcat:jmkj7tocuvhs5jbbl6qs4g25aa

Single pass sparsification in the streaming model with edge deletions [article]

Ashish Goel and Michael Kapralov and Ian Post
2012 arXiv   pre-print
[SODA'12] to estimate 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 .  ... 
arXiv:1203.4900v1 fatcat:qjjlfeaoonhsjgafpjnhw6aebq
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