Approximation Algorithms for Modern Multi-Processor Scheduling Problems

This thesis is devoted to the design and analysis of algorithms for scheduling problems. These problems are ubiquitous in the modern world. Examples include the optimization of local transportation, managing access to concurrent resources like runways at airports and efficient execution of computing tasks on server systems. Problem instances that appear in the real world often are so large and complex that it is not possible to solve them “by hand”. This rises the need for strong algorithmic approaches, which motivates our focus of study. In this work we consider two types of scheduling problems which gained in importance due to recent technological advances. The first problem comes from the avionics industry and deals with scheduling periodically recurring tasks in a parallel computer network on a plane: Each task comes with a period p and execution time c, and needs to use a processor exclusively for c time units every p time units. The scheduling problem is to assign starting offsets for the first execution of the tasks so that no collision occurs. The second problem is a scheduling problem that arises in highly parallelized processing environments with a shared common resource, e.g., modern multi-core computer architectures. In addition to classical makespan minimization problems such as scheduling on identical machines, each job has an additional resource constraint. The scheduler must ensure that at no time, the accumulated requirement of all active jobs at that time exceeds a given limit. For both types of problems we study their algorithmic complexity in a mathematical, rigorous way by designing approximation algorithms and establishing inapproximability results. We thereby give a characterization of the approximation landscape of these problems. We also consider a more practical perspective: For an engineer from the industry, a rigorous proof that an algorithm finds a solution of certain guaranteed quality for all possible kinds of problem instances is usually not that relevant. It is rather of interest to find “good enough” or even optimal solutions for particular instances that actually appear in the real world in “reasonable” time. We show that structural insights gained in the more theoretical process of designing approximation algorithms can be highly beneficial also for obtaining practical results. In particular, we develop integer programming formulations for the avionics problem based on structural properties revealed in the design of approximation algorithms. These formulations lead to strong tools that, for the first time, enable to algorithmically solve real-world instances from our industrial partner

Coloring Fuzzy Circular Interval Graphs

Given a graph G with nonnegative node labels w, a multiset of stable sets S1,..., Sk ⊆ V (G) such that each vertex v ∈ V (G) is contained in w(v) many of these stable sets is called a weighted coloring. The weighted coloring number χw(G) is the smallest k such that there exist stable sets as above. We provide a polynomial time combinatorial algorithm that computes the weighted coloring number and the corresponding colorings for fuzzy circular interval graphs. The algorithm reduces the problem to the case of circular interval graphs, then making use of a coloring algorithm by Gijswijt. We also show that the stable set polytopes of fuzzy circular interval graphs have the integer decomposition property