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Two Geometric Optimization Problems [chapter]

Bhaskar Dasgupta, Vwani Roychowdhury
1994 Nonconvex Optimization and Its Applications  
minimum number of rectangles included in the polygon."  ...  The rst one concerns the following minimization problem, termed as the rectilinear polygon cover problem: \Cover certain features of a given rectilinear polygon (possibly with rectilinear holes) with the  ...  Piotr Berman with whom part of the research work on this problem was carried out and Prof. Georg Schnitger for suggesting the problem of non-approximability o f t h e i n terior cover problem.  ... 
doi:10.1007/978-1-4613-3629-7_3 fatcat:5tfk57hc3jgkplngxnfxaymtoq

Jump Number of Two-Directional Orthogonal Ray Graphs [chapter]

José A. Soto, Claudio Telha
2011 Lecture Notes in Computer Science  
We model maximum cross-free matchings and minimum biclique covers of two-directional orthogonal ray graphs (2-dorgs) as maximum independent sets and minimum hitting sets of an associated family of rectangles  ...  For the subclass of convex graphs, our approach is a vast improvement over previous algorithms.  ...  Given a bipartite graph G = (V, E), a biclique is the edge set of a complete bipartite subgraph of G. A biclique cover is a family of bicliques whose union is E.  ... 
doi:10.1007/978-3-642-20807-2_31 fatcat:fn35p64m6jhkbepqrnpc2l47k4

Grid graphs and lattice surfaces [article]

W. Patrick Hooper
2012 arXiv   pre-print
First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to (m,n,∞) triangle groups.  ...  These surfaces were first discovered by Bouw and Möller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways.  ...  The author would especially like to thank John Smillie for realizing that a relatively simple presentation of these surfaces should be possible.  ... 
arXiv:0811.0799v3 fatcat:nxr7fuvonjhwjposnj7iw4e4ni

Grid Graphs and Lattice Surfaces

W. Patrick Hooper
2012 International mathematics research notices  
First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to (m, n, ∞) triangle groups.  ...  We construct these surfaces by gluing together polygons in two ways.  ...  The author would especially like to thank John Smillie for realizing that a relatively simple presentation of these surfaces should be possible.  ... 
doi:10.1093/imrn/rns124 fatcat:rprptpur2ja4latvwvxbmfrjfe

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon
2013 Annales Scientifiques de l'Ecole Normale Supérieure  
The phase space of the classical system contains, as an open dense subset, the moduli space L Γ of line bundles with connections on the graph Γ.  ...  We show that the dimer model on a bipartite graph Γ on a torus gives rise to a quantum integrable system of special type, which we call a cluster integrable system.  ...  He is grateful to Aspen Center of Physics and IHES for hospitality; R.K. is supported by the NSF.  ... 
doi:10.24033/asens.2201 fatcat:td6pyut3jjd5taelz6psl2rlqa

Dimers and cluster integrable systems [article]

A. B. Goncharov, R. Kenyon
2012 arXiv   pre-print
The sum of Hamiltonians is essentially the partition function of the dimer model. Any graph on a torus gives rise to a bipartite graph on the torus.  ...  The phase space of the classical system contains, as an open dense subset, the moduli space of line bundles with connections on the graph.  ...  He is grateful to Aspen Center of Physics and IHES for hospitality; R.K. is supported by the NSF.  ... 
arXiv:1107.5588v2 fatcat:452xsd3dmjgnzm7ng6rj4pq43y

Twenty Open Problems in Enumeration of Matchings [article]

James Propp
1999 arXiv   pre-print
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs.  ...  Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems in Enumeration of Matchings: Progress Report" (also available from this server as math.CO/9801061).  ...  polygonal cells that share an edge.  ... 
arXiv:math/9801060v3 fatcat:rzrdnvwcujdtndx6gk7dltfo6a

Combinatorial Bounds on Nonnegative Rank and Extended Formulations [article]

Samuel Fiorini and Volker Kaibel and Kanstantsin Pashkovich and Dirk Oliver Theis
2012 arXiv   pre-print
Extended formulations of small size (i.e., number of facets) are of interest, as they allow to model corresponding optimization problems as linear programs of small sizes.  ...  An extended formulation of a polytope P is a polytope Q which can be projected onto P.  ...  Rectangle covering is the same concept as covering the edges of a bipartite graph by bicliques (e.g., [8] ), and it also coincides with factorizing a Boolean matrix (e.g., [6, 16] ).  ... 
arXiv:1111.0444v2 fatcat:c7buz47olnhkhfljt5uyisvsde

Combinatorial bounds on nonnegative rank and extended formulations

Samuel Fiorini, Volker Kaibel, Kanstantsin Pashkovich, Dirk Oliver Theis
2013 Discrete Mathematics  
needed to cover the support of a slack matrix of the polytope.  ...  Our main goals are to shed some light on the question how this combinatorial rectangle covering bound compares to other bounds known from the literature, and to obtain a better idea of the power as well  ...  Acknowledgments We are grateful to the referees whose comments lead to significant improvements in the presentation of the material.  ... 
doi:10.1016/j.disc.2012.09.015 fatcat:iu3dejjm4vhmdh64ed2p6v4aoe

Unstable maps [article]

Gareth A. Jones
2018 arXiv   pre-print
The map is called stable if every automorphism of this cover is a lift of an automorphism of the map.  ...  This note describes several infinite families of unstable maps, and relates them to similar phenomena for graphs, hypermaps and Klein surfaces.  ...  Acknowledgement The author thanks David Singerman for a number of helpful comments.  ... 
arXiv:1810.02143v1 fatcat:r3zagf3qmnhznk7skixdog4cwm

Extended Formulations in Combinatorial Optimization [article]

Volker Kaibel
2011 arXiv   pre-print
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention  ...  In this paper (written for the newsletter Optima of the Mathematical Optimization Society), we provide a brief introduction to this topic and sketch some of the recent developments with respect to both  ...  One can equivalently describe the rectangle covering bound as the minimum number of complete bipartite subgraphs needed to cover the vertex-facet-nonincidence graph of the polytope P .  ... 
arXiv:1104.1023v1 fatcat:zeycppq4kjggpop5q6d67t2lhm

Efficient join-index-based spatial-join processing: a clustering approach

S. Shekhar, Chang-Tien Lu, S. Chawla, S. Ravada
2002 IEEE Transactions on Knowledge and Data Engineering  
The I/O cost of join computation using a join-index with limited buffer space depends primarily on the page-access sequence used to fetch the pages of the base relations.  ...  Given a join-index, we introduce a suite of methods based on clustering to compute the joins. We derive upper bounds on the length of the page-access sequences.  ...  ACKNOWLEDGMENTS This work is sponsored in part by the US Army High Performance Computing Research Center under the auspices of the Department of the Army, Army Research Laboratory cooperative agreement  ... 
doi:10.1109/tkde.2002.1047776 fatcat:ws5rfiz2gfeefanywedixvfjzq

Layout decomposition co-optimization for hybrid e-beam and multiple patterning lithography

Yunfeng Yang, Wai-Shing Luk, Hai Zhou, Changhao Yan, Xuan Zeng, Dian Zhou
2015 The 20th Asia and South Pacific Design Automation Conference  
In addition, a chain decomposition algorithm is employed for removing all "non-cyclable" edges. Furthermore, we investigate two versions of the PD method, one with planarization and one without.  ...  For K = 2, we propose a primaldual (PD) method for solving the underlying minimum odd-cycle cover problem efficiently.  ...  Yifang Chen from State Key Lab. of ASIC and System in Fudan University for providing the EBL system data and calculating the e-beam exposure time. We would like to thank Prof.  ... 
doi:10.1109/aspdac.2015.7059082 dblp:conf/aspdac/YangLZYZZ15 fatcat:ib2xkhas45gqlb7mxfhulzkvzy

Classifying grounded intersection graphs via ordered forbidden patterns [article]

Laurent Feuilloley, Michel Habib
2021 arXiv   pre-print
One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that are said to be grounded on a line.  ...  It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered  ...  This pattern correspond to Pb but with an edge instead of a non-edge, and it is also a permutation of the pattern for 3-colorable graph.  ... 
arXiv:2112.00629v2 fatcat:rpm3ou7qmvejjlc6lrfxt7yegq

Enumeration of Matchings: Problems and Progress [article]

James Propp
1999 arXiv   pre-print
I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article.  ...  The article concludes with a dozen new open problems. (Note: This article supersedes math.CO/9801060 and math.CO/9801061.)  ...  Acknowledgements This research was conducted with the support of the National Science Foundation, the National Security Agency, and the M.  ... 
arXiv:math/9904150v2 fatcat:y7o6jvl3ebb25mtdwzf73wzbkm
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