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Fixedparameter algorithms for protein similarity search under mRNA structure constraints
 In Proc. of the 31st international Workshop on Graphtheoretic concepts in computer science (WG
, 2005
"... Abstract. In the context of protein engineering, we consider the problem of computing an mRNA sequence of maximal codonwise similarity to a given mRNA (and consequently, to a given protein) that additionally satisfies some secondary structure constraints, the socalled mRNA Structure Optimization ( ..."
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Abstract. In the context of protein engineering, we consider the problem of computing an mRNA sequence of maximal codonwise similarity to a given mRNA (and consequently, to a given protein) that additionally satisfies some secondary structure constraints, the socalled mRNA Structure Optimization (MRSO) problem. Since MRSO is known to be APXhard, Bongartz [10] suggested to attack the problem using the approach of parameterized complexity. In this paper we propose three fixedparameter algorithms that apply for several interesting parameters of MRSO. We believe these algorithms to be relevant for practical applications today, as well as for possible future applications. Furthermore, our results extend the known tractability borderline of MRSO, and provide new research horizons for further improvements of this sort.
Fixed linear crossing minimization by reduction to the maximum cut problem
 in Proc 12th Ann. Int. Computing and Combinatorics Conference (COCOON’06
"... Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear ..."
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Abstract. Many reallife scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the socalled fixed linear crossing number problem (FLCNP). We show that this N Phard problem can be reduced to the wellknown maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branchandcut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branchandbound algorithms. 1
Crossing Minimization for Symmetries
 Proc. of ISAAC 2002, Lecture Notes in Computer Science
, 2002
"... We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NPhard, even if the order of orbits around the rotation center or along the reection axis is fixed. Nevertheless, there is a linear time algorithm to test ..."
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Cited by 5 (4 self)
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We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NPhard, even if the order of orbits around the rotation center or along the reection axis is fixed. Nevertheless, there is a linear time algorithm to test planarity and to construct a planar embedding if possible. Finally, we devise an O(m log m) algorithm for computing a crossing minimal drawing if interorbit edges may not cross orbits, showing in particular that intraorbit edges do not contribute to the NPhardness of the crossing minimization problem for symmetries.
Improved lower bounds for the 2page crossing numbers of Km,n and Kn via semidefinite programming
, 2011
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Line crossing minimization on metro maps
, 2007
"... We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying ..."
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Cited by 2 (1 self)
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We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway line which connects them, whereas the paths illustrate the lines connecting terminal stations. We call this the metroline crossing minimization problem (MLCM). In contrast to the problem of drawing the underlying graph nicely, MLCM has received fewer attention. It was recently introduced by Benkert et. al in [2]. In this paper, as a first step towards solving MLCM in arbitrary graphs, we study path and tree networks. We examine several variations of the problem for which we develop algorithms for obtaining optimal solutions.
An improved neural network model for the 2page crossing number problem
 IEEE TRANS. ON NEURAL NETWORKS
, 2006
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On MetroLine Crossing Minimization
, 2010
"... We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V, E) so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying n ..."
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We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V, E) so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway track connecting them, whereas the paths illustrate the metro lines connecting terminal stations. We call this the metroline crossing minimization problem (MLCM). We examine several variations of the problem for which we develop algorithms that yield optimal solutions.
Multilevel verticality optimization: Concept, strategies, and drawing scheme
, 2012
"... In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, an ..."
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In traditional multilevel graph drawing—known as Sugiyama’s framework—the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality in fact also have a low number of crossings. We present heuristic and exact approaches to tackle the verticality problem and study them in practice. Furthermore, we present a new drawing scheme (inherently bundling edges and drawing them monotonously), especially suitable for verticality optimization. It works without the traditional subdivision of edges, i.e., edges may span multiple levels, and therefore potentially allows to tackle larger graphs.
The Graph Crossing Number and its Variants: A Survey
"... The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introdu ..."
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The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems. 1 So, Which Crossing Number is it? The crossing number, cr(G), of a graph G is the smallest number of crossings required in any drawing of G. Or is it? According to a popular introductory textbook on combinatorics [320, page 40] the crossing number of a graph is “the minimum number of pairs of crossing edges in a depiction of G”. So, which one is it? Is there even a difference? To start with the second question, the easy answer is: yes, obviously there is a difference, the difference between counting all crossings and counting pairs of edges that cross. But maybe these different ways of counting don’t make a difference and always come out