Dagstuhl News January - December 2000

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Approximation hardness of deadline-TSP reoptimization

AbstractGiven an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e.g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i.e.,whether lm-U is computationally more tractable than U. While positive examples are known e.g. for metric TSP, we give some negative examples here: Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem

О несуществовании простого варианта полиномиального алгоритма извлечения корня из языка

For the usual operation of concatenation of words considered as multiplication, the concatenation of languages is obviously determined, and on the basis of the last operation, the degree of the language and the root of a given degree (if available) is determined. When describing algorithms for constructing a language that is a root of degree M from a given language, so called potential roots are of great importance: these are the words (not the languages) whose considered M-th degree is included in a given language. It is easily to show that all potential roots for a given language are constructed using a polynomial algorithm. This task, apparently, is not simplified when considering words and languages over the 1-letter alphabet, which is done in this paper. So called taboo pair of potential roots is a pair whose word concatenation is not included in the language. In previous publications on the topic of describing algorithms for extracting roots from a language, the hypothesis arose that a polynomial algorithm for extracting a root from a language can be described on the basis of considering the set of taboo pairs only, by iterating over specially described subsets of the set of potential roots. This paper shows, that such an algorithm (called “simple”) is impossible, i.e., if there is a polynomial algorithm for extracting the root from the language, then this algorithm must use some additional information.Для стандартной операции конкатенации слов, рассматриваемой как умножение, естественным образом определяется конкатенация языков, а на основе последней операции – степень языка и, при наличии, корень заданной степени. При описании алгоритмов построения языка, являющегося корнем степени M из заданного языка, большое значение имеют так называемые потенциальные корни: это такие слова (не языки), рассматриваемая M-я степень которых входит в заданный язык. Несложно показать, что все потенциальные корни для заданного языка строятся с помощью полиномиального алгоритма. Эта задача, по-видимому, не упрощается при рассмотрении слов и языков над 1-буквенным алфавитом — что и делается в настоящей статье. Табуированная пара потенциальных корней — это такая пара, конкатенация слов которой в язык не входит. В предыдущих публикациях на тему описания алгоритмов извлечения корней из языка возникала гипотеза, что полиномиальный алгоритм извлечения корня из языка может быть описан на основе рассмотрения только множества табуированных пар — путем перебора специально описываемых подмножеств множества потенциальных корней. В настоящей статье показывается, что подобный алгоритм (называемый «простым») невозможен, т.е. если и существует полиномиальный алгоритм извлечения корня из языка, то он (алгоритм) должен использовать некоторую дополнительную информацию

On the Approximability of TSP on Local Modifications of Optimally Solved Instances

Given an instance of TSP together with an optimal solution, we consider the scenario in which this instance is modified locally, where a local modification consists in the alteration of the weight of a single edge. More generally, for a problem U, let LM-U (local-modification-U) denote the same problem as U, but in LM-U, we are also given an optimal solution to an instance from which the input instance can be derived by a local modification. The question is how to exploit this additional knowledge, i.e., how to devise better algorithms for LM-U than for U. Note that this need not be possible in all cases: The general problem of LM-TSP is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β-triangle inequality (LM-Δβ-TSP) remains NP-hard for all β>½. However, for LM-Δ-TSP (i.e., metric LM-TSP), we will present an efficient 1.4-approximation algorithm. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. Similarly, for all 1<β<3.34899, we achieve a better approximation ratio for LM-Δ-TSP than for Δβ-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs

Approximation Algorithms for Steiner Tree Problems Based on Universal Solution Frameworks

This paper summarizes the work on implementing few solutions for the Steiner Tree problem which we undertook in the PAAL project. The main focus of the project is the development of generic implementations of approximation algorithms together with universal solution frameworks. In particular, we have implemented Zelikovsky 11/6-approximation using local search framework, and 1.39-approximation by Byrka et al. using iterative rounding framework. These two algorithms are experimentally compared with greedy 2-approximation, with exact but exponential time Dreyfus-Wagner algorithm, as well as with results given by a state-of-the-art local search techniques by Uchoa and Werneck. The results of this paper are twofold. On one hand, we demonstrate that high level algorithmic concepts can be designed and efficiently used in C++. On the other hand, we show that the above algorithms with good theoretical guarantees, give decent results in practice, but are inferior to state-of-the-art heuristical approaches

Constructing Incremental Sequences in Graphs

Given a weighted graph , we investigate the problem of constructing a sequence of subsets of vertices (called groups) with small diameters, where the diameter of a group is calculated using distances in G. The constraint on these n groups is that they must be incremental: . The cost of a sequence is the maximum ratio between the diameter of each group Mi and the diameter of a group with I vertices and minimum diameter: . This quantity captures the impact of the incremental constraint on the diameters of the groups in a sequence. We give general bounds on the value of this ratio and we prove that the problem of constructing an optimal incremental sequence cannot be solved approximately in polynomial time with an approximation ratio less than 2 unless P = NP. Finally, we give a 4-approximation algorithm and we show that the analysis of our algorithm is tight