Dirk Bongartz

On the hardness of constructing minimal 2-connected spanning subgraphs in complete graphs with sharpened triangle inequality

In this paper we investigate the problem of finding a 2-connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, for both the edge and the vertex connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [1, 1 + e], for an arbitrary small e > 0. This result implies the first explicit lower bound on the approximability of the general version (i.e., for arbitrary graphs) of the problem. On the other hand, if the input graph satisfies the sharpened β-triangle inequality, then a (2/3 + 1/3 ċ β/1-β)-approximation algorithm is designed. This ratio tends to 1 with β tending to ½, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as β < 5/7Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened β-triangle inequality with β < 2/3, then this ratio can be improved to β/1-β.

On the Hardness of Constructing Minimal 2-Connected Spanning Subgraphs in Complete Graphs with Sharpened Triangle Inequality

In this paper we investigate the problem of finding a 2- connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, both for the edge- and for the vertex-connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval [1,1 + ?], for an arbitrary small ? > 0. This result implies the first explicit lower bound on the approximability of the general problems.On the other hand, if the input graph satisfies the sharpened s-triangle inequality, then a (2/3 + 1/3, s/1-s)-approximation algorithm is designed. This ratio tends to 1 with s tending to 1/2, and it improves the previous known bound of 3/2, holding for graphs satisfying the triangle inequality, as soon as s < 5/7.Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened s-triangle inequality with s < 2/3, then this ratio can be improved to s/1-s.

Protein folding in the HP model on grid lattices with diagonals

The protein folding problem, i.e., the computational prediction of the three-dimensional structure of a protein from its amino acid sequence, is one of the most important and challenging problems in computational biology. Since a complete simulation of the folding process of a protein is far too complex to handle, one tries to find an approximate solution by using a simplified, abstract model. One of the most popular models is the so-called HP model, where the hydrophobic interactions between the amino acids are considered to be the main force in the folding process, and furthermore the folding space is modeled by a two- or three-dimensional grid lattice. In this paper, we will present some approximation algorithms for the protein folding problem in the HP model on an extended grid lattice with plane diagonals. The choice of this kind of lattice removes one of the major drawbacks of the original HP model, namely the bipartiteness of the grid which severely restricts the set of possible foldings. Our algorithms achieve an approximation ratio of 2615~1.733 for the two-dimensional and of 85=1.6 for the three-dimensional lattice. This improves significantly over the best previously known approximation ratios for the protein folding problem in the HP model on any lattice.

Some Notes on the Complexity of Protein Similarity Search under mRNA Structure Constraints

In [2], Backofen et al. propose the MRSO problem, that is to compute an mRNA sequence of maximal similarity to a given mRNA and a given protein, that additionally satisfies some secondary structure constraints. The study of this problem is motivated by an application in the area of protein engineering. Modeled in a mathematical framework, we would like to compute a string \({s\in\{a,b,\overline{a},\overline{b}\}}^{3n}\) which maximizes the sum of the values of n functions, which are blockwise applied to triples of s, and additionally satisfies some complementary constraints on the characters of s given in terms of position pairs. While the decision version of this problem is known to be NP-complete (see [2]), we prove here the APX-hardness of the general as well as of a restricted version of the problem. Moreover, we attack the problem by proposing a 4-approximation algorithm.

On k-edge-connectivity problems with sharpened triangle inequality

The edge-connectivityproblem is to find a minimum-cost k-edge-connected spanning subgraph of an edge-weighted, undirected graph G for any given G and k. Here we consider its APX-hard subproblems with respect to the parameter β, with 1/2 ≤ β < 1, where G = (V, E) is a complete graph with a cost function c satisfying the sharpened triangle inequality c({u, v}) ≤ β ċ (c({u, w}) + c({w, v})) for all u, v, w ∈ V. First, we give a linear-time approximation algorithm for these optimization problems with approximation ratio β/1-β for any 1/2 ≤ β < 1, which does not depend on k. The result above is based on a rough combinatorial argumentation. We sophisticate our combinatorial consideration in order to design a (1 + 5(2β-1)/9(1-β))- approximation algorithm for the 3-edge-connectivitysubgraph problem for graphs satisfying the sharpened triangle inequality for 1/2 ≤ β ≤ 2/3.

Public-Key Cryptography

This chapter presents an asymmetric approach to cryptography – it uses different keys for encryption and decryption. The key for encryption is announced publically so that everybody can encode a message. Only the owner of the matching secret key can decode the encrypted message. Most cryptographic schemes used today in the Internet are based on asymmetric algorithms using a secret and a public key.

Public-Key-Kryptographie

Wer wollte nicht schon mal eine Geheimnachricht ubermitteln? Sogar Casar hat das schon gemacht. Angeblich verschob er einfach jeden Buchstaben seiner Nachricht im Alphabet um drei Positionen weiter nach rechts. Aus einem A wird bei diesem Verfahren ein D, aus einem B ein E usw. und schlieslich aus einem W ein Z, aus einem X ein A, aus einem Y ein B und aus einem Z ein C.

A weighted HP model for protein folding with diagonal contacts

The HP model is one of the most popular discretized models for attacking the protein folding problem, i.e. , for the computational prediction of the tertiary structure of a protein from its amino acid sequence. It is based on the assumption that interactions between hydrophobic amino acids are the main force in the folding process. Therefore, it distinguishes between polar and hydrophobic amino acids only and tries to embed the amino acid sequence into a two- or three-dimensional grid lattice such as to maximize the number of contacts, i.e. , of pairs of hydrophobic amino acids that are embedded into neighboring positions of the grid. In this paper, we propose a new generalization of the HP model which overcomes one of the major drawbacks of the original HP model, namely the bipartiteness of the underlying grid structure which severely restricts the set of possible contacts. Moreover, we introduce the (biologically well-motivated) concept of weighted contacts, where each contact gets assigned a weight depending on the spatial distance between the embedded amino acids. We analyze the applicability of existing approximation algorithms for the original HP model to our new setting and design a new approximation algorithm for this generalized model.

Grundlagen der Molekularbiologie

Wenn man wie in diesem Buch Fragestellungen im Bereich der Molekularbiologie behandeln will, so ist zur Entwicklung und Bewertung der abstrakten Modelle und Techniken die Kenntnis zumindest grundlegender Zusammenhange in der Molekularbiologie erforderlich. Wir widmen daher dieses Kapitel dem Basiswissen uber die Biomolekule, die wir im weiteren Verlauf des Buches betrachten werden. Die Darstellungen der Sachverhalte sind dabei wiederum Abstraktionen und Zusammenfassungen aller molekularbiologischen Kenntnisse, sie erheben weder Anspruch auf Vollstandigkeit noch auf Detailgenauigkeit. Sie sollen dem Leser einen Uberblick uber die grundlegenden Zusammenhange geben, so dass die nachfolgend in diesem Buch beschriebenen Problemstellungen auf einer soliden Datenbasis von Seiten der Molekularbiologie fusen. Dementsprechend werden wir auf die Details verzichten, die fur das Verstandnis des weiteren Textes nicht unbedingt erforderlich sind.

Bestimmung der Basensequenz

In diesem Kapitel werden wir uns mit dem Vorgang der Sequenzierung von DNA bzw. von Nucleotidsequenzen im Allgemeinen beschaftigen. Darunter verstehen wir die Bestimmung der Basenabfolge in einem Nucleinsauremolekul. Im Folgenden werden wir exemplarisch die DNA als ein solches Nucleinsauremolekul betrachten.