TY  - JOUR
A1  - Buchin, Kevin
A1  - Buchin, Maike
A1  - Byrka, Jaroslaw
A1  - Nöllenburg, Martin
A1  - Okamoto, Yoshio
A1  - Silveira, Rodrigo I.
A1  - Wolff, Alexander
T1  - Drawing (Complete) Binary Tanglegrams
JF  - Algorithmica
N2  - A binary tanglegram is a drawing of a pair of rooted binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example, in phylogenetics, it is essential that both trees are drawn without edge crossings and that the inter-tree edges have as few crossings as possible. It is known that finding a tanglegram with the minimum number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number.

We prove that under the Unique Games Conjecture there is no constant-factor approximation for binary trees. We show that the problem is NP-hard even if both trees are complete binary trees. For this case we give an O(n 3)-time 2-approximation and a new, simple fixed-parameter algorithm. We show that the maximization version of the dual problem for binary trees can be reduced to a version of MaxCut for which the algorithm of Goemans and Williamson yields a 0.878-approximation.
KW  - NP-hardness
KW  - crossing minimization
KW  - binary tanglegram
KW  - approximation algorithm
KW  - fixed-parameter tractability
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-124622
VL  - 62
ER  - 
TY  - JOUR
A1  - Atienza, Nieves
A1  - de Castro, Natalia
A1  - Cortés, Carmen
A1  - Garrido, M. Ángeles
A1  - Grima, Clara I.
A1  - Hernández, Gregorio
A1  - Márquez, Alberto
A1  - Moreno-González, Auxiliadora
A1  - Nöllenburg, Martin
A1  - Portillo, José Ramón
A1  - Reyes, Pedro
A1  - Valenzuela, Jesús
A1  - Trinidad Villar, Maria
A1  - Wolff, Alexander
T1  - Cover contact graphs
N2  - We study problems that arise in the context of covering certain geometric objects called seeds (e.g., points or disks) by a set of other geometric objects called cover (e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, respectively, but they can touch. We call the contact graph of a cover a cover contact graph (CCG). We are interested in three types of tasks, both in the general case and in the special case of seeds on a line: (a) deciding whether a given seed set has a connected CCG, (b) deciding whether a given graph has a realization as a CCG on a given seed set, and (c) bounding the sizes of certain classes of CCG’s. Concerning (a) we give efficient algorithms for the case that seeds are points and show that the problem becomes hard if seeds and covers are disks. Concerning (b) we show that this problem is hard even for point seeds and disk covers (given a fixed correspondence between graph vertices and seeds). Concerning (c) we obtain upper and lower bounds on the number of CCG’s for point seeds.
KW  - Informatik
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-78845
ER  -