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A tighter insertionbased approximation of the crossing number
, 2011
"... Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the ..."
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Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NPhard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the Falmostplanar graph G+F, while computing the crossing number of G+F exactly is NPhard already when F = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of Falmostplanar graphs, achieving constantfactor approximation for the large class of such graphs of bounded degrees and bounded size of F.
Advances in the Planarization Method: Effective Multiple Edge Insertions
"... Abstract. The planarization method is the strongest known method to heuristically find good solutions to the general crossing number problem in graphs: starting from a planar subgraph, one iteratively inserts edges, representing crossings via dummy nodes. In the recent years, several improvements bo ..."
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Abstract. The planarization method is the strongest known method to heuristically find good solutions to the general crossing number problem in graphs: starting from a planar subgraph, one iteratively inserts edges, representing crossings via dummy nodes. In the recent years, several improvements both from the practical and the theoretical point of view have been made. We review these advances and conduct an extensive study of the algorithms ’ practical implications. Thereby, we present the first implementation of an approximation algorithm for the crossing number problem of general graphs, and compare the obtained results with known exact crossing number solutions. 1
Acknowledgements
, 2006
"... Finding behavioral mechanisms and ecological patterns in sparse and noisy data ..."
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Finding behavioral mechanisms and ecological patterns in sparse and noisy data
A Combinatorial Approach to Orthogonal . . .
, 2001
"... We study two families of NPhard orthogonal placement problems that arise in the area of information visualization both from a theoretical and a practical point of view. This thesis contains a common combinatorial framework for compaction problems in orthogonal graph drawing and for pointfeature la ..."
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We study two families of NPhard orthogonal placement problems that arise in the area of information visualization both from a theoretical and a practical point of view. This thesis contains a common combinatorial framework for compaction problems in orthogonal graph drawing and for pointfeature labeling problems in computational cartography. Compaction problems are concerned with performing the conversion from a dimensionless description of the orthogonal shape of a graph to an areaefficient drawing in the orthogonal grid with short edges. The second family of problems deals with the task of attaching rectangular labels to pointfeatures such as cities or mountain peaks on a map so that the placement results in a legible map. We present new combinatorial formulations for these problems employing a path and cyclebased graphtheoretic property in an associated problemspecific pair of constraint graphs. The reformulation allows us to develop exact algorithms for the original problems. Extensive computational studies on realworld benchmarks show that our linear programming–based algorithms are able to solve large instances of the placement problems to provable optimality within short computation time. Furthermore,