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Page 4722 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
are planar graphs almost as good as the complete graphs and almost as cheap as minimum spanning trees.  ...  by 2r+ 1 times the length of a minimum Euclidean spanning tree on S.  ... 

A PTAS for planar group Steiner tree via spanner bootstrapping and prize collecting

MohammadHossein Bateni, Erik D. Demaine, MohammadTaghi Hajiaghayi, Dániel Marx
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016  
Spanner bootstrapping removes one of the main barriers for designing PTASs for problems which have no known constant-factor approximation (even on planar graphs), and thus can be used to obtain PTASs for  ...  Our second major contribution required for the planar group Steiner tree PTAS is a spanner construction, which *  ...  The next step of the algorithm finally reaches our main goal: the groups have no tips, that is, every terminal is on the prespanner.  ... 
doi:10.1145/2897518.2897549 dblp:conf/stoc/BateniDHM16 fatcat:gd2tnsws5rdyxc7orfzxzopxwy

Combinatorial Problems in Solving Linear Systems [chapter]

Iain Duff, Bora Uçar
2012 Combinatorial Scientific Computing  
Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise.  ...  On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid.  ...  Amestoy and Jean-Yves L'Excellent for their contribution to the presented material. We also thankÜmit V. Ç atalyürek for his comments on an earlier version of the paper.  ... 
doi:10.1201/b11644-3 fatcat:wdq7hb343jawxjaqsxz5nihtfq

A Practical Algorithm for the Computation of the Genus [article]

G. Brinkmann
2020 arXiv   pre-print
Apart from the specific problem and the results, the article can also be seen as an example how some design principles used to carefully develop and implement standard backtracking algorithms can still  ...  We describe a practical algorithm to compute the (oriented) genus of a graph, give results of the program implementing this algorithm, and compare the performance to existing algorithms.  ...  If one wanted to apply the algorithm to perform very well when testing only planarity, one could apply the reasoning of Demoucron, Malgrange and Pertuiset [8] as soon as a spanning tree and the first  ... 
arXiv:2005.08243v1 fatcat:eiibn3s44zctle4mp223tehbvq

The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree [article]

Jakub Łącki and Jakub Oćwieja and Marcin Pilipczuk and Piotr Sankowski and Anna Zych
2016 arXiv   pre-print
We consider the model where we are given an n-vertex graph G=(V,E,w) with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning  ...  For planar graphs we achieve the same running time and the approximation ratio of (2+ε). Moreover, we show faster algorithms for incremental and decremental scenarios.  ...  Since the space usage is O(τ −1 n log n log D), the copying requires O(τ −2 n log n log(n/τ ) log D)) = O(ε −2 n log n log(n/ε) log D)) time, and so is the space usage.  ... 
arXiv:1308.3336v5 fatcat:d5s44uqxd5ctlodlozixlzrake

The Power of Dynamic Distance Oracles

Jakub Łącki, Jakub Oćwieja, Marcin Pilipczuk, Piotr Sankowski, Anna Zych
2015 Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing - STOC '15  
We consider the model where we are given an n-vertex graph G = (V, E, w) with positive real edge weights, and our goal is to maintain a tree which is a good approximation of the minimum Steiner tree spanning  ...  For planar graphs we achieve the same running time and the approximation ratio of (2 + ε). Moreover, we show faster algorithms for incremental and decremental scenarios.  ...  Since the space usage is O(τ −1 n log n log D), the copying requires O(τ −2 n log n log(n/τ ) log D)) = O(ε −2 n log n log(n/ε) log D)) time, and so is the space usage.  ... 
doi:10.1145/2746539.2746615 dblp:conf/stoc/LackiOPSZ15 fatcat:bo7oiwea7ne43ooq5ho4ycw5by

Graph Profiling for Vertex Cover: Targeted Reductions in a Branch and Reduce Solver [article]

Matthias F. Stallmann, Yang Ho, Timothy D. Goodrich
2020 arXiv   pre-print
Our research question is are there graph characteristics that determine which reductions will be most effective? Not only is the answer affirmative, but relevant characteristics are easy to identify.  ...  Our primary contributions are 1. A thorough examination reduction routine performance in the context of graph characteristics. 2.  ...  The authors thank Yoichi Iwata for his help navigating some of the details of VCSolver, and for providing insights about the LP reduction that allowed us to create a stand-alone implementation.  ... 
arXiv:2003.06639v1 fatcat:yyer6jxjqfacff3fj3pqdyv7cy

Electrical Flows over Spanning Trees [article]

Swati Gupta, Ali Khodabakhsh, Hassan Mortagy, Evdokia Nikolova
2020 arXiv   pre-print
been developed in the power systems community as early as 1989.  ...  The tree requirement on the support of the flow is motivated by operational constraints in electricity distribution networks.  ...  grid embedding of planar graphs, and the anonymous referees for numerous useful suggestions to improve this manuscript.  ... 
arXiv:1909.04759v3 fatcat:hlzwwbnhwjf6ppmjpsuiedx6l4

On Problems without Polynomial Kernels (Extended Abstract) [chapter]

Hans L. Bodlaender, Rodney G. Downey, Michael R. Fellows, Danny Hermelin
2008 Lecture Notes in Computer Science  
These problems include k- Path, k-Cycle, k-Exact Cycle, k-Short Cheap Tour, k-Graph Minor Order Test, k-Cutwidth, k-Search Number, k-Pathwidth, k-Treewidth, k-Branchwidth, and several optimization problems  ...  Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NP-hard problems.  ...  The fourth author would also like to thank Moritz Müller for reviewing several preliminary versions, and especially for the countless (and sometimes endless) debates on related topics.  ... 
doi:10.1007/978-3-540-70575-8_46 fatcat:sd3fuidkzvay7hvljo257xpvjy

Approximating Node-Weighted k-MST on Planar Graphs [article]

Jarosław Byrka, Mateusz Lewandowski, Joachim Spoerhase
2018 arXiv   pre-print
We study the problem of finding a minimum weight connected subgraph spanning at least k vertices on planar, node-weighted graphs. We give a (4+)-approximation algorithm for this problem.  ...  We achieve this by utilizing the recent LMP primal-dual 3-approximation for the node-weighted prize-collecting Steiner tree problem by Byrka et al (SWAT'16) and adopting an approach by Chudak et al.  ...  Acknowledgements We would like to thank Zachary Friggstad for initial discussions on the problem. The authors were supported by the NCN grant number 2015/18/E/ST6/00456.  ... 
arXiv:1801.00313v2 fatcat:caqusdn6jbceto25yphdq45zva

An Introduction to Temporal Graphs: An Algorithmic Perspective [article]

Othon Michail
2015 arXiv   pre-print
When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence G_1,G_2...  ...  Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs.  ...  For example, the first labeled problem introduced in the literature was the Labeled Minimum Spanning Tree problem, which has several applications in communication network design.  ... 
arXiv:1503.00278v1 fatcat:iwdk4wqfhzfafnfqke7wer66nu

Approximation Algorithms for NP-Hard Problems

Ravindran Kannan, Marek Karpinski
2004 Oberwolfach Reports  
The Program of the meeting and Abstracts of all talks are listed in the subsequent sections of this report.  ...  In addition to 25 lectures delivered at general sessions, there were several additional lectures given at the special sessions and the evening problem session.  ...  Acknowledgements This research was supported by The Israel Science Foundation (grant No. 263/02).  ... 
doi:10.4171/owr/2004/28 fatcat:fwbs36pgpjev5gk6cfeb7ylukm

The Laplacian on planar graphs and graphs on surfaces

Richard Kenyon
2011 Current Developments in Mathematics  
In particular we review and extend results on the reconstruction of the network from boundary measurements, connections with the spanning tree and grove model, and the characteristic polynomial of the  ...  We discuss the Laplacian operator on graphs embedded on a surface, concentrating on simple surfaces (disk, annulus, torus).  ...  This completes the proof. The spanning tree measure is the probability measure on spanning trees of a graph in which the probability of a tree is proportional to its weight.  ... 
doi:10.4310/cdm.2011.v2011.n1.a1 fatcat:zs4aibc2n5fwvnquu3ehmxmt6a

The Laplacian on planar graphs and graphs on surfaces [article]

Richard Kenyon
2012 arXiv   pre-print
The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdi\'ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree  ...  We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface.  ...  This completes the proof. The spanning tree measure is the probability measure on spanning trees of a graph in which the probability of a tree is proportional to its weight.  ... 
arXiv:1203.1256v1 fatcat:6ppapkqdwvhohjooy6ej4god34

Vertex-Disjoint Packing of Two Steiner Trees: polyhedra and branch-and-cut

Eduardo Uchoa, Marcus Poggi de Aragão
2001 Mathematical programming  
The resulting code is capable of solving problems on grid graphs with up to 10000 vertices and 5000 terminals in a few minutes.  ...  A realistic model for this problem is given by the vertex-disjoint packing of two Steiner trees (2VPST), which is known to be NP-complete. This work presents an investigation on the 2VPST polyhedra.  ...  Such policy is e cient because p-crosses are (i) very cheap to separate over planar graphs, (ii) very sparse, therefore they do not burden the LPs and (iii) very strong cuts in practice. 2-separators  ... 
doi:10.1007/pl00011434 fatcat:jin5dvmz2fbptf5sjgmjsfx7ga
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