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Vertex Connectivity in Poly-logarithmic Max-flows [article]

Jason Li, Danupon Nanongkai, Debmalya Panigrahi, Thatchaphol Saranurak, Sorrachai Yingchareonthawornchai
2021 arXiv   pre-print
In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances.  ...  Using this reduction, we can solve vertex connectivity in Õ(m^α) time for any α≥ 1, if there is a m^α-time maxflow algorithm.  ...  Panigrahi has been supported in part by NSF Awards CCF 1750140 and CCF 1955703.  ... 
arXiv:2104.00104v2 fatcat:jj7f7t6jfff5nkzy6yb2pupyi4

Set connectivity problems in undirected graphs and the directed steiner network problem

Chandra Chekuri, Guy Even, Anupam Gupta, Danny Segev
2011 ACM Transactions on Algorithms  
We describe the first poly-logarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O(log 2 n log 2 k).  ...  of disjoint vertex subsets.  ...  A Poly-Logarithmic Approximation for Generalized Connectivity In what follows, we present a poly-logarithmic approximation for the generalized connectivity problem.  ... 
doi:10.1145/1921659.1921664 fatcat:sb2mxwwshzbsvbaxsjyqmvglha

Long Range Percolation Mixing Time [article]

Itai Benjamini, Noam Berger, Ariel Yadin
2009 arXiv   pre-print
While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic  ...  We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ).  ...  Remarks • In the following sections we will provide upper and lower bounds on the mixing time of G s,β (N ), that match up to poly-logarithmic factors.  ... 
arXiv:math/0703872v2 fatcat:rma6gvde2zfzvdiiuugg7gm3gy

Approximability of Capacitated Network Design [chapter]

Deeparnab Chakrabarty, Chandra Chekuri, Sanjeev Khanna, Nitish Korula
2011 Lecture Notes in Computer Science  
We also consider a variant of the Cap-SNDP in which multiple copies of an edge can be bought: we give an O(log k) approximation for this case, where k is the number of vertex pairs with non-zero connectivity  ...  We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s, t) has positive connectivity requirement) captures much of the difficulty of Cap-SNDP:  ...  CC's interest in capacitated network design was inspired by questions from Matthew Andrews.  ... 
doi:10.1007/978-3-642-20807-2_7 fatcat:cxduegjs2ffl5knfecxzk4kss4

Approximability of Capacitated Network Design

Deeparnab Chakrabarty, Chandra Chekuri, Sanjeev Khanna, Nitish Korula
2014 Algorithmica  
We also consider a variant of the Cap-SNDP in which multiple copies of an edge can be bought: we give an O(log k) approximation for this case, where k is the number of vertex pairs with non-zero connectivity  ...  We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s, t) has positive connectivity requirement) captures much of the difficulty of Cap-SNDP:  ...  CC's interest in capacitated network design was inspired by questions from Matthew Andrews.  ... 
doi:10.1007/s00453-013-9862-4 fatcat:m4iu23ts4ndyrkilrobdm6cvvi

Cut-Matching Games on Directed Graphs [article]

Anand Louis
2010 arXiv   pre-print
Our algorithm uses only O(log^2 n) single commodity max-flow computations and thus breaks the multicommodity-flow barrier for computing the sparsest cut on directed graphs  ...  This framework reduces the NP-hard DSC problem to the computation of a poly-logarithmic number of single-commodity max-flows while, at the same time, keeping the approximation factor poly-logarithmic.  ...  On a different direction of study, Arora and Kale [5] gave an algorithm that achieves an approximation ratio of O(log n) while still running in time dominated by poly-logarithmic single commodity max-flow  ... 
arXiv:1010.1047v1 fatcat:ay2zxixeb5ee7oc5vejoercdge

Deterministic Tree Embeddings with Copies for Algorithms Against Adaptive Adversaries [article]

Bernhard Haeupler and D Ellis Hershkowitz and Goran Zuzic
2021 arXiv   pre-print
In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing O(log n) copies of each vertex while preserving the connectivity  ...  Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms.  ...  is poly-logarithmic in the total number of possible revealed elements).  ... 
arXiv:2102.05168v1 fatcat:sbpdn7bupvbb7epghfqcma4ic4

Deterministic Min-cut in Poly-logarithmic Max-flows [article]

Jason Li, Debmalya Panigrahi
2022 arXiv   pre-print
This marks the first improvement for this problem since a running time bound of Õ(mn) was established by several papers in the early 1990s.  ...  Given a set of vertices R, this entails finding cuts of minimum weight that separate (or isolate) each individual vertex v∈ R from the rest of the vertices R∖{v}.  ...  Acknowledgements JL was supported in part by NSF award CCF-1907820. DP was supported in part by NSF award CCF-1955703 and an NSF CAREER award CCF-1750140.  ... 
arXiv:2111.02008v2 fatcat:2d74kzhpzzddro6c4srwf6nn5i

Approximation Algorithms for Orienting Mixed Graphs [chapter]

Michael Elberfeld, Danny Segev, Colin R. Davidson, Dana Silverbush, Roded Sharan
2011 Lecture Notes in Computer Science  
Our algorithm provides a sub-linear guarantee in the general case, and logarithmic guarantees for structured instances.  ...  When the input graph is undirected, a sub-logarithmic approximation is known for the problem.  ...  a poly-logarithmic fraction of the input pairs.  ... 
doi:10.1007/978-3-642-21458-5_35 fatcat:oefqitkwnrfcfc5cuzvzbbwivu

Stackelberg Network Pricing Games

Patrick Briest, Martin Hoefer, Piotr Krysta
2010 Algorithmica  
Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex cover, which is based on non-trivial max-flow and LP-duality techniques  ...  We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m pricable edges in a graph. The other edges have a fixed cost.  ...  In bipartite graphs the problem can be solved optimally by using a classic and fundamental max-flow/min-cut argumentation.  ... 
doi:10.1007/s00453-010-9480-3 fatcat:dd5b4mu73vbjrc2phmx6ovqlty

Approximation algorithms for orienting mixed graphs

Michael Elberfeld, Danny Segev, Colin R. Davidson, Dana Silverbush, Roded Sharan
2013 Theoretical Computer Science  
Our algorithm provides a sub-linear guarantee in the general case, and logarithmic guarantees for structured instances.  ...  When the input graph is undirected, a sub-logarithmic approximation is known for the problem.  ...  a poly-logarithmic fraction of the input pairs.  ... 
doi:10.1016/j.tcs.2012.03.044 fatcat:xnlp26mjlnaermoav5xzh3wl4i

A sufficiently fast algorithm for finding close to optimal clique trees

Ann Becker, Dan Geiger
2001 Artificial Intelligence  
α is the approximation ratio of an α-approximation algorithm for the 3-way vertex cut problem.  ...  When α = 4/3, our algorithm's complexity is O(2 4.67k n • poly(n)) and it errs by a factor of 3.67 where poly(n) is the running time of linear programming.  ...  The conference version of this paper can be found in the Proceedings of the 12th conference on Uncertainty in Artificial Intelligence.  ... 
doi:10.1016/s0004-3702(00)00075-8 fatcat:nqoekroozjagbehhirvh6mi5ku

A Sufficiently Fast Algorithm for Finding Close to Optimal Junction Trees [article]

Ann Becker, Dan Geiger
2013 arXiv   pre-print
The algorithm guarantees that the logarithm of the size of the state space of the heaviest clique in the junction tree produced is less than a constant factor off the optimal value.  ...  The algorithm has a worst case complexity O(c^k n^a) where a and c are constants, n is the number of vertices, and k is the size of the largest clique in a junction tree of G in which this size is minimized  ...  Finding a minjmum vertex cut is done by any of the well known max-flow I min cut algorithms.  ... 
arXiv:1302.3558v1 fatcat:chvprsz7wndvfeg4ndzw3bhtne

Approximation Algorithms for Non-Uniform Buy-at-Bulk Network Design

C. Chekuri, M. Hajiaghayi, G. Kortsarz, M. Salavatipour
2006 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06)  
We improve upon this result, by presenting the first poly-logarithmic approximation for this problem.  ...  The ratio we obtain is O(log 3 h · min{log D, γ(h 2 )}) where h is the number of pairs and γ(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over  ...  As a byproduct of Theorem 1.1 we can obtain such an algorithm where α and β are poly-logarithmic in h.  ... 
doi:10.1109/focs.2006.15 dblp:conf/focs/ChekuriHKS06 fatcat:wtpha4pnqngyrc3plbol4akkim

Approximation Algorithms for Nonuniform Buy-at-Bulk Network Design

C. Chekuri, M. T. Hajiaghayi, G. Kortsarz, M. R. Salavatipour
2010 SIAM journal on computing (Print)  
We improve upon this result, by presenting the first poly-logarithmic approximation for this problem.  ...  The ratio we obtain is O(log 3 h · min{log D, γ(h 2 )}) where h is the number of pairs and γ(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over  ...  As a byproduct of Theorem 1.1 we can obtain such an algorithm where α and β are poly-logarithmic in h.  ... 
doi:10.1137/090750317 fatcat:3cuyzk2fnrgwhgqbe4yafjriri
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