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Online Bipartite Matching with Amortized O(^2 n) Replacements [article]

Aaron Bernstein, Jacob Holm, Eva Rotenberg
2018 arXiv   pre-print
In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges  ...  The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching.  ...  This motivates the online bipartite matching problem with replacements, where the goal is to at all times match as many clients as possible, while minimizing the number of changes to the matching.  ... 
arXiv:1707.06063v4 fatcat:vge4e6tkvrdfvprzgz7uwkdnby

Online Bipartite Matching with Amortized O(log 2 n) Replacements

Aaron Bernstein, Jacob Holm, Eva Rotenberg
2019 Journal of the ACM  
In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges  ...  The problem of online bipartite matchings with replacements was introduced in 1995 by Grove, Kao, Krishnan, and Vitter [13], who showed matching upper and lower bounds of Θ(n log n) replacements for the  ...  the amortized number of replacements.  ... 
doi:10.1145/3344999 fatcat:2krkte564jh4bheagk23jdvhx4

Theoretical Foundations of Storage Systems (Dagstuhl Seminar 19111)

Martin Farach-Colton, Inge Li Gørtz, Rob Johnson, Donald E. Porter, Michael Wagner
2019 Dagstuhl Reports  
This seminar brought together researchers from two distinct communities -algorithms researchers with an interest in external memory and systems researchers with an interest in storage -with the objective  ...  Online Bipartite Matching with Amortized O(log 2 n) Replacements Eva Rotenberg (DTU -Copenhagen, DK) License Creative Commons BY 3.0 Unported license © Eva Rotenberg Joint work of Aaron Bernstein, Jacob  ...  Holmm, Eva Rotenberg In this talk, we introduce the online with recourse model, and give an example: online bipartite matchings.  ... 
doi:10.4230/dagrep.9.3.39 dblp:journals/dagstuhl-reports/Farach-ColtonGJ19 fatcat:pfbfw7sgzjeq7eoqnjurmwzq5u

Trade-offs in dynamic coloring for bipartite and general graphs [article]

Manas Jyoti Kashyop, N. S. Narayanaswamy, Meghana Nasre, Sai Mohith Potluri
2020 arXiv   pre-print
We present a deterministic fully dynamic 2-coloring algorithm with O(log^2 n) amortized update time, O(log n) amortized query time, and one recoloring in the worst case.  ...  We then show a deterministic incremental 2-coloring algorithm which does not maintain color of every vertex after each update, with amortized O(α(n)) update time, amortized O(α(n)) recolorings, and amortized  ...  and deletion in amortized O(γ + log n) update time and worst case O(1) query time.  ... 
arXiv:1909.07854v3 fatcat:6mcwi2nnmvgplh5elgkl5ff4hm

On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter [article]

Søren Dahlgaard
2016 arXiv   pre-print
Based on popular conjectures we show that: -- No algorithm with amortized update time O(n^1-ε) exists for incremental or decremental maximum cardinality bipartite matching.  ...  Furthermore no algorithm with amortized update time O(n^1-ε) exists for directed and unweighted graphs or undirected and weighted graphs. -- No algorithm with amortized update time O(n^1/2 - ε) exists  ...  I would also like to thank an anonymous referee for pointing out the reduction from incremental matching to undirected flow.  ... 
arXiv:1602.06705v2 fatcat:os3gntola5bcnea5wl67dxdjpa

A better tester for bipartiteness? [article]

Andrej Bogdanov, Fan Li
2010 arXiv   pre-print
We conjecture that the induced subgraph is Ω (ϵ)-far from bipartite with high probability.  ...  Discrete Math. 15(2): 211-227 (2002)) show that if a graph is ϵ-far from bipartite, then the subgraph induced by a random subset of O(1/ϵ) vertices is bipartite with high probability.  ...  The idea of Reduce is to build (in an online manner) an expander Z on the dense components, while replacing the sparse components with loops.  ... 
arXiv:1011.0531v1 fatcat:b53457abqfeornwrh4hbbgexya

Optimal Point Movement for Covering Circular Regions [article]

Danny Z. Chen and Xuehou Tan and Haitao Wang and Gangshan Wu
2011 arXiv   pre-print
A by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm can handle each vertex insertion or deletion on the graph  ...  For the min-sum problem, we show that a special case with all points initially lying on the boundary of the circular region can be solved in O(n^2) time, improving a previous O(n^4) time solution.  ...  Yet, the correctness of our (online) dynamic maximum matching algorithm for circular convex bipartite graphs can be seen quite easily.  ... 
arXiv:1107.1012v1 fatcat:b472zaxdkncldlszshrfgbn5sq

Online Minimum Cost Matching with Recourse on the Line

Nicole Megow, Lukas Nölke, Raghu Meka, Jarosław Byrka
2020 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We show an O(1)-competitive algorithm for online matching on the line with amortized recourse of O(log n). This is the first non-trivial result for min-cost bipartite matching with recourse.  ...  In online minimum cost matching on the line, n requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line.  ...  The online bipartite matching problem on the line admits an O(1)-competitive algorithm with amortized recourse budget O(log n).  ... 
doi:10.4230/lipics.approx/random.2020.37 dblp:conf/approx/MegowN20 fatcat:j7bfejhu75fippc4invdpw6coq

Optimal Point Movement for Covering Circular Regions [chapter]

Danny Z. Chen, Xuehou Tan, Haitao Wang, Gangshan Wu
2012 Lecture Notes in Computer Science  
In addition, a by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; Algorithmica (2015) 72:379-399 our algorithm can handle  ...  For the min-sum problem we show that a special case with all points initially lying on the boundary of the circular region can be solved in O(n 2 ) time, improving a previous O(n 4 ) time solution.  ...  Yet, the correctness of our (online) dynamic maximum matching algorithm for circular convex bipartite graphs can be seen quite easily.  ... 
doi:10.1007/978-3-642-35261-4_36 fatcat:6tbb72xq3jgqtfu5fphqqhgr5q

Optimal Point Movement for Covering Circular Regions

Danny Z. Chen, Xuehou Tan, Haitao Wang, Gangshan Wu
2013 Algorithmica  
In addition, a by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; Algorithmica (2015) 72:379-399 our algorithm can handle  ...  For the min-sum problem we show that a special case with all points initially lying on the boundary of the circular region can be solved in O(n 2 ) time, improving a previous O(n 4 ) time solution.  ...  Yet, the correctness of our (online) dynamic maximum matching algorithm for circular convex bipartite graphs can be seen quite easily.  ... 
doi:10.1007/s00453-013-9857-1 fatcat:wkgw4s3wkbbkfhmeplaslfpftm

(1 + ∊)-Approximate Incremental Matching in Constant Deterministic Amortized Time [chapter]

Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn,, Shay Solomon
2019 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms  
We present a deterministic algorithm that, for any constant ε > 0, maintains a (1 + ε)approximate matching with constant amortized update time per insertion.  ...  We study the matching problem in the incremental setting, where we are given a sequence of edge insertions and aim at maintaining a near-maximum cardinality matching of the graph with small update time  ...  The only previous result for approximate matchings in the incremental model is due to Gupta [16] , who gave an amortized O(log 2 n) update-time algorithm to maintain (1 + ε)-approximate matchings in bipartite  ... 
doi:10.1137/1.9781611975482.114 dblp:conf/soda/0001LSSS19 fatcat:u4n2qalic5gzrhjjcqby66gx6e

(1 + eps)-Approximate Incremental Matching in Constant Deterministic Amortized Time

Fabrizio Grandoni, Stefano Leonardi, Piotr Sankowski, Chris Schwiegelshohn, Shay Solomon
2020 Zenodo  
We present a deterministic algorithm that, for any constant \(\varepsilon>0\), maintains a \((1+\varepsilon)\)-approximate matching with constant amortized update time per insertion.  ...  We study the matching problem in the incremental setting, where we are given a sequence of edge insertions and aim at maintaining a near-maximum cardinality matching of the graph with small update time  ...  The only previous result for approximate matchings in the incremental model is due to Gupta [16] , who gave an amortized O(log 2 n) update-time algorithm to maintain (1 + ε)-approximate matchings in bipartite  ... 
doi:10.5281/zenodo.4586902 fatcat:fcgckh737nfypcoo3mh5u4rttm

Online Minimum Cost Matching on the Line with Recourse [article]

Nicole Megow University of Bremen, Germany)
2020 arXiv   pre-print
We show an O(1)-competitive algorithm for online matching on the line that uses at most O(nlog n) reassignments. This is the first non-trivial result for min-cost bipartite matching with recourse.  ...  In online minimum cost matching on the line, n requests appear one by one and have to be matched immediately and irrevocably to a given set of servers, all on the real line.  ...  The online bipartite matching problem on the line admits a constant competitive algorithm with amortized recourse budget O(n log n).  ... 
arXiv:2001.03107v1 fatcat:w6v5oawkzrgc7pbu7meaegi6ia

Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms [article]

Amir Abboud, Søren Dahlgaard
2016 arXiv   pre-print
Using our framework, we show that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized O(n^1/2-ε) time, for ε>  ...  An algorithm with O(polylog(n)) runtime would be a major breakthrough.  ...  No algorithm can solve the dynamic maximum weight matching problem in bipartite planar graphs on N nodes with amortized update time upN q and query time qpN q such that maxpqpN q, upN qq " OpN 1 2´ε q  ... 
arXiv:1605.03797v1 fatcat:nvi2uvrd25dbrhwg6wgz5x6tsu

Mind the Gap [article]

Amihood Amir, Tsvi Kopelowitz, Avivit Levy, Seth Pettie, Ely Porat, B. Riva Shalom
2015 arXiv   pre-print
We examine the complexity of the online Dictionary Matching with One Gap Problem (DMOG) which is the following.  ...  In more general versions the gap symbols are associated with bounds determining the possible lengths of matching strings.  ...  While dictionary matching is well studied (see, e.g. [2, 4, 11, 5, 14] ), NIDS applications motivate the dictionary matching with one gap problem, defined formally as follows. Definition 1.  ... 
arXiv:1503.07563v2 fatcat:fuy2e5ftenbazme22dsluitohu
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