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Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
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The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
Drawing (complete) binary tanglegrams: Hardness, approximation, fixedparameter tractability
, 2008
"... A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertree edges ..."
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Cited by 9 (2 self)
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A binary tanglegram is a pair 〈S, T 〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn without edge crossings and that the intertree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NPhard and that the problem is fixedparameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constantfactor approximation for general binary trees. We show that the problem is hard even if both trees are complete binary trees. For this case we give an O(n³)time 2approximation and a new and simple fixedparameter algorithm. We show that the maximization version of the dual problem for general binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878approximation.
Matched Drawings of Planar Graphs
, 2007
"... A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique ycoordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs all ..."
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Cited by 7 (3 self)
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A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique ycoordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3connected planar graphs or a 3connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a planar graph of some special families—such as unlabeled level planar (ULP) graphs or the family of “carousel graphs”—are always matched drawable.
Drawing Binary Tanglegrams: An Experimental Evaluation
, 2009
"... A tanglegram is a pair of trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. In applications such as phylogenetics or hierarchical clustering, it is required that the individual trees are drawn crossingfree. A natural optimization problem, deno ..."
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A tanglegram is a pair of trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. In applications such as phylogenetics or hierarchical clustering, it is required that the individual trees are drawn crossingfree. A natural optimization problem, denoted tanglegram layout problem, is thus to minimize the number of crossings between intertree edges. The tanglegram layout problem is NPhard even for complete binary trees, for general binary trees the problem is hard to approximate if the Unique Games Conjecture holds. In this paper we present an extensive experimental comparison of a new and several known heuristics for the general binary case. We measure the performance of the heuristics with a simple integer linear program and a new exact branchandbound algorithm. The new heuristic returns the first solution that the branchandbound algorithm computes (in quadratic time). Surprisingly, in most cases this simple heuristic is at least as good as the best of the other heuristics.
Untangling Tanglegrams: Comparing Trees by their Drawings ∗
, 2009
"... A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimi ..."
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A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimizations problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was also studied by Fernau, Kauffman and Poths (FSTTCS 2005). A similar reduction to a graph crossing problem also helps to solve an open problem they posed, showing a fixedparameter tractable algorithm for minimizing the number of crossings over all dary trees. For the case where one tree is fixed, we show an O(n log n) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman’s footrule distance and give an O(n 2) algorithm. We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber (Austral. Symp. on Info. Vis. 2004) to minimize crossings. 1
Generalized binary tanglegrams: Algorithms and applications
 In Proc. of BICOP 2009, volume 5462 of LNCS
, 2009
"... Abstract. Several applications require the joint display of two phylogenetic trees whose leaves are matched by intertree edges. This issue arises, for example, when comparing gene trees and species trees or when studying the cospeciation of hosts and parasites. The tanglegram layout problem seeks ..."
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Abstract. Several applications require the joint display of two phylogenetic trees whose leaves are matched by intertree edges. This issue arises, for example, when comparing gene trees and species trees or when studying the cospeciation of hosts and parasites. The tanglegram layout problem seeks to produce a layout of the two trees that minimizes the number of crossings between the intertree edges. This problem is wellstudied for the case when the mappings between the leaves of the two trees is onetoone. However, in typical biological applications, this mapping is seldom onetoone. In this work we (i) define a generalization of the tanglegram layout problem, called the Generalized Tanglegram Layout (GTL) problem, which allows for arbitrary interconnections between the leaves of the two trees, (ii) provide efficient algorithms for the case when the layout of one tree is fixed, (iii) discuss the fixed parameter tractability and approximability of the GTL problem, (iv) formulate heuristic solutions for the GTL problem, and (v) evaluate our algorithms experimentally. 1
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
Parameterized Algorithms for dHitting Set: the Weighted Case
, 2008
"... We are going to analyze simple search tree algorithms for Weighted dHitting Set. Although the algorithms are simple, their analysis is technically rather involved. However, this approach allows us to even improve on elsewhere previously published running time estimates for the more restricted cas ..."
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We are going to analyze simple search tree algorithms for Weighted dHitting Set. Although the algorithms are simple, their analysis is technically rather involved. However, this approach allows us to even improve on elsewhere previously published running time estimates for the more restricted case of (unweighted) dHitting Set.
Drawing binary tanglegrams: Hardness, approximation, fixedparameter tractability
"... A binary tanglegram is a pair 〈S, T〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn with no edge crossing and that the intertree edges hav ..."
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Cited by 1 (1 self)
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A binary tanglegram is a pair 〈S, T〉 of binary trees whose leaf sets are in onetoone correspondence; matching leaves are connected by intertree edges. For applications, for example in phylogenetics, it is essential that both trees are drawn with no edge crossing and that the intertree edges have as few crossings as possible. It is known that finding a drawing with the minimum number of crossings is NPhard and that the problem is fixedparameter tractable with respect to that number. We show that the problem is hard even if both trees are complete binary trees. For this case we give an O(n³)time 2approximation and a new and simple fixedparameter algorithm. We prove that under the Unique Games Conjecture there is no constantfactor approximation for general binary trees. We show that the maximization version of the problem for general binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878approximation.