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Vertex Sparsifiers for c-Edge Connectivity
[article]
2019
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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]
2010
Lecture Notes in Computer Science
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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]
2022
arXiv
pre-print
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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
2012
Proceedings of the 44th symposium on Theory of Computing - STOC '12
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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]
2012
arXiv
pre-print
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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
2016
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
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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]
2014
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
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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]
2014
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2021
arXiv
pre-print
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(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
2015
Proceedings of the 34th ACM Symposium on Principles of Database Systems - PODS '15
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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]
2012
arXiv
pre-print
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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]
2012
arXiv
pre-print
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[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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