Approximation Algorithms for Scheduling with Reservations

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan Cmax, which is the maximum completion time. The general case of the problem is inapproximable unless P = NP; hence we study a suitable strongly NP-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemeted by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless P = NP. The PTAS presented here is the rst one for the problem under consideration; so far, not even for well-know special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a vonstant approximation ratio and discuss FPTAes for special cases as well as the complexity of the problem if m is part of the input

The Diversity of Religious Diversity. Using Census and NCS Methodology in Order to Map and Assess the Religious Diversity of a Whole Country

Religious diversity is often captured in “mapping studies” that use mostly qualitative methods in order to map and assess the religious communities in a given area. While these studies are useful, they often present weaknesses in that they treat only limited geographic regions, provide limited possibilities for comparing across religious groups and cannot test theories. In this article, we show how a census and a quantitative national congregations study (NCS) methodology can be combined in order to map and assess the religious diversity of a whole country (Switzerland), overcoming the problems mentioned above. We outline the methodological steps and selected results concerning organizational, geographic, structural, and cultural diversity

Approximability of Scheduling with Fixed Jobs

The scheduling problem of minimizing the makespan is among the most well studied problems --- especially in the field of approximation. In modern industrial software however, it has become standard to work on a variant of this problem, where some of the jobs are already fixed in the schedule. The remaining jobs are to be assigned to the machines in such a way that they do not overlap with fixed jobs. In this paper we first focus on simple algorithms for this problem which have a reasonable performance guarantee and are easy to implement in practical settings. This is followed by a polynomial time approximation scheme (PTAS) for the case that the number m of machines is constant. Moreover, we show that, unless P = NP, there is no arbitrarily close approximation in the general case when the number of machines is part of the input. This will be extended by showing that there is no asymptotic PTAS in the general machine case. We finally show that there exists no FPTAS in the constant machine case, unless P = NP

The thickness of a minor-excluded class of graphs

The thickness problem on graphs is $\cal NP$-hard and only few results concerning this graph invariant are known. Using a decomposition theorem of Truemper, we show that the thickness of the class of graphs without $G_{12}$-minors is less than or equal to two (and therefore, the same is true for the more well-known class of the graphs without $K_5$-minors). Consequently, the thickness of this class of graphs can be determined with a planarity testing algorithm in linear time

The thickness of graphs: a survey

We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs