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Shortest Augmenting Paths for Online Matchings on Trees

Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, Anna Zych-Pawlewicz
2018 Theory of Computing Systems  
The shortest augmenting path (SAP) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J.  ...  In this paper we prove an O(n log 2 n) upper bound for the total length of augmenting paths for trees.  ...  Shortest Paths on Trees In this section we study the shortest augmenting path (SAP) algorithm, which in each turn chooses the shortest among all available augmenting paths.  ... 
doi:10.1007/s00224-017-9838-x fatcat:zmvpmb4bqrg4bld4gcrjnpah7y

WAOA 2015 Special Issue on TOCS

Laura Sanità, Martin Skutella
2018 Theory of Computing Systems  
paths found by the shortest augmenting path algorithm for online matchings on trees.  ...  In their article Shortest Augmenting Paths for Online Matchings on Trees, Bartlomiej Bosek, Dariusz Leniowski, Piotr Sankowski, and Anna Zych-Pawlewicz, give a new bound on the total length of all augmenting  ... 
doi:10.1007/s00224-018-9850-9 fatcat:ulo7cobavfelzjctlackzjne7e

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

Aaron Bernstein, Jacob Holm, Eva Rotenberg
2018 arXiv   pre-print
We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(^2 n) replacements per insertion, where  ...  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  ...  for very helpful pointers regarding the details of this proof.  ... 
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  
We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized O(log 2 n) replacements per insertion,  ...  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  ...  Consider the following protocol for constructing a matching: For each client c in arbitrary order, augment along the shortest augmenting path from c (if one exists).  ... 
doi:10.1145/3344999 fatcat:2krkte564jh4bheagk23jdvhx4

A Tight Bound for Shortest Augmenting Paths on Trees [article]

Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, Anna Zych-Pawlewicz
2017 arXiv   pre-print
Shortest augmenting paths for online matchings on trees. In WAOA 2015].  ...  The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms.  ...  By the statement (ii) of Lemma 5 we have that there is a matching for which path t (b t0 ) is one of the shortest augmenting path.  ... 
arXiv:1704.02093v2 fatcat:o7mul3wxtzdorg7c2sssvfmmcq

A Tight Bound for Shortest Augmenting Paths on Trees [chapter]

Bartłomiej Bosek, Dariusz Leniowski, Piotr Sankowski, Anna Zych-Pawlewicz
2018 Lecture Notes in Computer Science  
The shortest augmenting path technique is one of the fundamental ideas used in maximum matching and maximum flow algorithms.  ...  In this paper we prove a tight O(n log n) upper bound for the total length of shortest augmenting paths for trees improving over O(n log 2 n) bound [BLSZ15].  ...  Our work is meant as a step forward towards understanding the shortest augmenting path method for computing the matching on bipartite graphs.  ... 
doi:10.1007/978-3-319-77404-6_16 fatcat:6po4ykynvre5jeifw2p354bheq

Robustification of Online Graph Exploration Methods [article]

Franziska Eberle, Alexander Lindermayr, Nicole Megow, Lukas Nölke, Jens Schlöter
2021 arXiv   pre-print
We initiate the study of a learning-augmented variant of the classical, notoriously hard online graph exploration problem by adding access to machine-learned predictions.  ...  By interpolating carefully between a given algorithm and NN, we prove new performance bounds that leverage the individual good performance on particular inputs while establishing robustness to arbitrary  ...  Here, a matching lower bound is attained on a weighted path. Only for rather special graph classes it is known how to obtain constant-competitive tours.  ... 
arXiv:2112.05422v1 fatcat:4vj243qg6jfsli6asq2w6xfit4

Achieving Competitiveness in Online Problems [article]

Mustafa Safa Ozdayi
2020 arXiv   pre-print
In addition to that, we analyze one of the algorithms, the shortest path algorithm, against an adversary. Through that, we prove some new results about algorithms performance.  ...  We then consider the online bipartite b-matching. In this problem, we are given a bipartite graph where one side of the graph is initially present and where the other side arrive online.  ...  Each node in T is matched to 2K nodes in L. Hence, the total number of matches of T is simply |T | · 2K. Shortest Augmenting Path Algorithm Consider the arrival of a node v L .  ... 
arXiv:2001.03015v1 fatcat:pkui65j4dncntn3udcwohnskba

Algorithms for the Masses

Robert Sedgewick
2011 2011 Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
performance of Ford-Fulkerson implementations • shortest augmenting path • maximum-capacity augmenting path Graph parameters for a reasonable model V E C upper bound shortest VE/2 VC  ...  of Ford-Fulkerson implementations • shortest augmenting path • maximum-capacity augmenting path Graph parameters for a reasonable model V = 177 E = 2000 C = 100 upper bound for example actual  ... 
doi:10.1137/1.9781611973013.6 dblp:conf/analco/Sedgewick11 fatcat:rok2c6bo6fgo5carfrzcfb32ka

Learning-Augmented Algorithms for Online Steiner Tree [article]

Chenyang Xu, Benjamin Moseley
2021 arXiv   pre-print
We consider the online Steiner tree problem in this model for both directed and undirected graphs.  ...  We show on graphs where terminals are drawn from a distribution, the new online algorithms have strong performance even with modestly correct predictions.  ...  We thank Yuyan Wang for sharing their experimental data and thank the anonymous reviewers for their insightful comments and suggestions.  ... 
arXiv:2112.05353v1 fatcat:gai5ciysfffeja7jkr3erqctfm

On Correcting Inputs: Inverse Optimization for Online Structured Prediction [article]

Hal Daumé III, Samir Khuller, Manish Purohit, Gregory Sanders
2015 arXiv   pre-print
We consider such inverse optimization problems for maximum weight matroid basis, matroid intersection, perfect matchings, minimum cost maximum flows, and shortest paths and derive the first known results  ...  The effectiveness of these algorithmic approaches to online learning for structured prediction is also discussed.  ...  The inverse shortest path tree problem is then to minimally modify the edge weights so that T spt becomes the δ-optimal shortest path tree, i.e., for every vertex v( = r) in G, the path prescribed by T  ... 
arXiv:1510.03130v1 fatcat:lo67kwu3obanfbo2fkpbweff2u

Robustification of Online Graph Exploration Methods

Franziska Eberle, Alexander Lindermayr, Nicole Megow, Lukas Nölke, Jens Schlöter
2022 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We initiate the study of a learning-augmented variant of the classical, notoriously hard online graph exploration problem by adding access to machine-learned predictions.  ...  By interpolating carefully between a given algorithm and NN, we prove new performance bounds that leverage the individual good performance on particular inputs while establishing robustness to arbitrary  ...  Here, a matching lower bound is attained on a weighted path. Only for rather special graph classes it is known how to obtain constant-competitive tours.  ... 
doi:10.1609/aaai.v36i9.21208 fatcat:iyxva6bfkzgsvk3lyrwkzrsy4y

Learning-Augmented Algorithms for Online Steiner Tree

Chenyang Xu, Benjamin Moseley
2022 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We consider the online Steiner tree problem in this model for both directed and undirected graphs.  ...  We show on graphs where terminals are drawn from a distribution, the new online algorithms have strong performance even with modestly correct predictions.  ...  We thank Yuyan Wang for sharing their experimental data and thank the anonymous reviewers for their insightful comments and suggestions.  ... 
doi:10.1609/aaai.v36i8.20854 fatcat:r6x4je5tnza3thbw6wabplhwca

Grafalgo - A Library of Graph Algorithms and Supporting Data Structures (revised) [article]

Jonathan Turner
2016 arXiv   pre-print
The readability of the code also makes it relatively straightforward to extend it for other purposes.  ...  The library is available on an open-source basis and may be downloaded from https://code.google.com/p/grafalgo/. Source code documentation is at www.arl.wustl.edu/ jst/doc/grafalgo.  ...  Grafalgo includes several algorithms for finding matchings in bipartite graphs, based on the concept of augmenting paths.  ... 
arXiv:1601.01597v1 fatcat:tnnsmnbupje3fjte2efi2zmls4

Algorithms for Matching and Predicting Trajectories [chapter]

Jochen Eisner, Stefan Funke, Andre Herbst, Andreas Spillner, Sabine Storandt
2011 2011 Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments (ALENEX)  
Related Work In many applications, map matching is an online problem, that is, location measurements have to be aligned with the road map as soon as they are taken (see e.g. [11] for more on this version  ...  We consider the following two problems: Map Matching: Given a sequence of (imprecise) location measurements from a mobile user moving on a road network, determine the most likely path in the network this  ...  The next time we leave this shortest path tree, we again start a search for such a suffix.  ... 
doi:10.1137/1.9781611972917.9 dblp:conf/alenex/EisnerFHSS11 fatcat:ynatkfd6qjbxnjnoogu3k4q3ia
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