Andreas Sprock

The string guessing problem as a method to prove lower bounds on the advice complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

Advice Complexity of the Online Coloring Problem

We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively.

The String Guessing Problem as a Method to Prove Lower Bounds on the Advice Complexity

The advice complexity of an online problem describes the additional information both necessary and sufficient for online algorithms to compute solutions of a certain quality. In this model, an oracle inspects the input before it is processed by an online algorithm. Depending on the input string, the oracle prepares an advice bit string that is accessed sequentially by the algorithm. The number of advice bits that are read to achieve some specific solution quality can then serve as a fine-grained complexity measure. The main contribution of this paper is to study a powerful method for proving lower bounds on the number of advice bits necessary. To this end, we consider the string guessing problem as a generic online problem and show a lower bound on the number of advice bits needed to obtain a good solution. We use special reductions from string guessing to improve the best known lower bound for the online set cover problem and to give a lower bound on the advice complexity of the online maximum clique problem.

On the Hardness of Reoptimization with Multiple Given Solutions

In reoptimization, we consider the following scenario: Given an instance of a hard optimization problem together with an optimal solution for it, we want to solve a locally modified instance of the problem. It has recently been shown for several hard optimization problems that their corresponding reoptimization variants remain NP-hard or even hard to approximate whereas they often admit improved approximation ratios. In this paper, we investigate a generalization of the reoptimization concept where we are given not only one optimal solution but multiple optimal solutions for an instance. We prove, for some variants of the Steiner tree problem and the traveling salesman problem, that the known reoptimization hardness results carry over to this generalized setting. Moreover, we consider the performance of local search strategies on reoptimization problems. We show that local search does not work for solving TSP reoptimization, even in the presence of multiple solutions.

The steiner tree reoptimization problem with sharpened triangle inequality

In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened β-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-costs are restricted to the values 1 and 1+γ for an arbitrary small γ>0), Steiner tree reoptimization still is NP-hard for several different types of local modifications, and even APX-hard for some of them. As for the upper bounds, for some local modifications, we design linear-time (1/2+β)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (β=1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2β-approximation algorithm for the classical Steiner tree problem on such instances, which might be of independent interest since it improves over the previously best known ratio for any β<1/2+ln (3)/4≈0.775.

Steiner tree reoptimization in graphs with sharpened triangle inequality

In this paper, we deal with several reoptimization variants of the Steiner tree problem in graphs obeying a sharpened @b-triangle inequality. A reoptimization algorithm exploits the knowledge of an optimal solution to a problem instance for finding good solutions for a locally modified instance. We show that, in graphs satisfying a sharpened triangle inequality (and even in graphs where edge-costs are restricted to the values 1 and 1+@c for an arbitrary small @c>0), Steiner tree reoptimization still is NP-hard for several different types of local modifications, and even APX-hard for some of them. As for the upper bounds, for some local modifications, we design linear-time (1/2+@b)-approximation algorithms, and even polynomial-time approximation schemes, whereas for metric graphs (@b=1), none of these reoptimization variants is known to permit a PTAS. As a building block for some of these algorithms, we employ a 2@b-approximation algorithm for the classical Steiner tree problem on such instances, which might be of independent interest since it improves over the previously best known ratio for any @b<1/2+ln(3)/4~0.775.

Teaching Public-Key Cryptography in School

These days, public-key cryptography is indispensable to ensure both confidentiality and authenticity in numerous applications which comprise securely communicating via mobile phone or email or digitally signing documents. For all public-key systems, such as RSA, mathematically challenging and technically involved methods are employed which are often above the level of secondary school students as they employ deep results from algebra. Following an approach suggested in 2003 by Tim Bell et al. in Computers and Education, volume 40, number 3 , we deal with the question of how to teach young students the main concepts, issues, and solutions of public-key systems without being forced to also teach rather complicated theorems of number theory beforehand.