### Refine

#### Keywords

- Informatik (1)
- NP-hard (1)
- NP-hardness (1)
- RNA-sequencing (1)
- approximation algorithm (1)
- binary tanglegram (1)
- cartographic requirements (1)
- colorectal cancer (CRC) (1)
- crossing minimization (1)
- fixed-parameter tractability (1)

In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments.
We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs.
We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s{t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution.

Cover contact graphs
(2012)

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.

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.

Beyond maximum independent set: an extended integer programming formulation for point labeling
(2017)

Map labeling is a classical problem of cartography that has frequently been approached by combinatorial optimization. Given a set of features in a map and for each feature a set of label candidates, a common problem is to select an independent set of labels (that is, a labeling without label–label intersections) that contains as many labels as possible and at most one label for each feature. To obtain solutions of high cartographic quality, the labels can be weighted and one can maximize the total weight (rather than the number) of the selected labels. We argue, however, that when maximizing the weight of the labeling, the influences of labels on other labels are insufficiently addressed. Furthermore, in a maximum-weight labeling, the labels tend to be densely packed and thus the map background can be occluded too much. We propose extensions of an existing model to overcome these limitations. Since even without our extensions the problem is NP-hard, we cannot hope for an efficient exact algorithm for the problem. Therefore, we present a formalization of our model as an integer linear program (ILP). This allows us to compute optimal solutions in reasonable time, which we demonstrate both for randomly generated point sets and an existing data set of cities. Moreover, a relaxation of our ILP allows for a simple and efficient heuristic, which yielded near-optimal solutions for our instances.

Background:
Colorectal cancer (CRC) is the second leading cause of cancer-related death in men and women. Systemic disease with metastatic spread to distant sites such as the liver reduces the survival rate considerably. The aim of this study was to investigate the changes in gene expression that occur on invasion and expansion of CRC cells when forming metastases in the liver.
Methods:
The livers of syngeneic C57BL/6NCrl mice were inoculated with 1 million CRC cells (CMT-93) via the portal vein, leading to the stable formation of metastases within 4 weeks. RNA sequencing performed on the Illumina platform was employed to evaluate the expression profiles of more than 14,000 genes, utilizing the RNA of the cell line cells and liver metastases as well as from corresponding tumour-free liver.
Results:
A total of 3329 differentially expressed genes (DEGs) were identified when cultured CMT-93 cells propagated as metastases in the liver. Hierarchical clustering on heat maps demonstrated the clear changes in gene expression of CMT-93 cells on propagation in the liver. Gene ontology analysis determined inflammation, angiogenesis, and signal transduction as the top three relevant biological processes involved. Using a selection list, matrix metallopeptidases 2, 7, and 9, wnt inhibitory factor, and chemokine receptor 4 were the top five significantly dysregulated genes.
Conclusion:
Bioinformatics assists in elucidating the factors and processes involved in CRC liver metastasis. Our results support the notion of an invasion-metastasis cascade involving CRC cells forming metastases on successful invasion and expansion within the liver. Furthermore, we identified a gene expression signature correlating strongly with invasiveness and migration. Our findings may guide future research on novel therapeutic targets in the treatment of CRC liver metastasis.