Sebastian Stiller

The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications

We present a new concept for optimization under uncertainty: recoverable robustness. A solution is recovery robust if it can be recovered by limited means in all likely scenarios. Specializing the general concept to linear programming we can show that recoverable robustness combines the flexibility of stochastic programming with the tractability and performances guarantee of the classical robust approach. We exemplify recoverable robustness in delay resistant, periodic and aperiodic timetabling problems, and train platforming.

Feasibility Analysis in the Sporadic DAG Task Model

Real-time systems increasingly contain processing units with multiple cores. To use this additional computational power in hard deadline environments, one needs schedulability tests for task models that represent the possibilities of parallel execution of jobs of a task. A standard model is to represent a (sporadically) recurrent task by a directed a cyclic graph (DAG). The nodes of the DAG correspond to the jobs of the task. All such jobs are released simultaneously, have to be completed within some common relative deadline, and some pairs of jobs are linked by a precedence constraint, i.e., an arc of the DAG. This poses new challenges for analyzing whether a task system is feasible, in particular for the commonly used online algorithms Earliest Deadline First (EDF) and Deadline Monotonic (DM). While for ordinary sporadic tasks the required algorithmic techniques are well-understood, despite recent research much remains open in this model. In this work, we completely close the gap between the algorithmic understanding of feasibility analysis for the usual sporadic task model and the case where each sporadic task is a DAG. We show for DAG tasks that EDF has a tight speedup bound of 2 - 1/m, where m is the number of processors, while DM has a speedup bound of at most 3 - 1/m. Moreover, we present polynomial and pseudopolynomial time tests, of differing effectiveness, for determining whether a set of sporadic DAG tasks can be scheduled by EDF or DM to meet all deadlines on a specified number of processors. We remark that the effectiveness of some of our tests matches the best known algorithms for ordinary sporadic task sets, thus closing the gap.A model has been proposed in [1] for representing recurrent precedence-constrained tasks to be executed on multiprocessor platforms, where each recurrent task is modeled by a directed acyclic graph (DAG), a period, and a relative deadline. Each vertex of the DAG represents a sequential job, while the edges of the DAG represent precedence constraints between these jobs. All the jobs of the DAG are released simultaneously and have to be completed within some specified relative deadline. The task may release jobs in this manner an unbounded number of times, with successive releases occurring at least the specified period apart. The feasibility problem is to determine whether such a recurrent task can be scheduled to always meet all deadlines on a specified number of dedicated processors. The case of a single task has been considered in [1]. The main contribution of this paper is to consider the case of multiple tasks. We show that EDF has a speedup bound of 2 − 1/m, where m is the number of processors. Moreover, we present polynomial and pseudopolynomial schedulability tests, of differing effectiveness, for determining whether a set of sporadic DAG tasks can be scheduled by EDF to meet all deadlines on a specified number of processors.

Delay resistant timetabling

In public transport punctuality has prominent influence on the customers’ satisfaction. Our task is to support a management decision to optimally invest passengers’ nominal travel time to secure the nominal schedule against delay. For aperiodic scheduling we clarify the notion and use of a fixed amount of time supplements, so-called buffers, both theoretically and by realistic examples. The general tool to solve such optimization problems is a sampling approach. We show how this approach is mathematically justified. As its applicability to large networks is limited, we show an efficient alternative for the case of series-parallel graphs. For periodic timetabling we propose two heuristic approaches to ensure a certain level of delay resistance at the least expense of slightly increased nominal passengers travel time, and analyze in detail their advantages and drawbacks.

Online railway delay management: Hardness, simulation and computation

Delays in a railway network are a common problem that railway companies face in their daily operations. When a train is delayed, it may either be beneficial to let a connecting train wait so that passengers in the delayed train do not miss their connection, or it may be beneficial to let the connecting train depart on time to avoid further delays. These decisions naturally depend on the global structure of the network, on the schedule, on the passenger routes and on the imposed delays. The railway delay management (RDM) problem (in a broad sense) is to decide which trains have to wait for connecting trains and which trains have to depart on time. The offline version (i.e. when all delays are known in advance) is already NP-hard for very special networks. In this paper we show that the online railway delay management (ORDM) problem is PSPACE-hard. This result justifies the need for a simulation approach to evaluate wait policies for ORDM. For this purpose we present TOPSU—RDM, a simulation platform for evaluating and comparing different heuristics for the ORDM problem with stochastic delays. Our novel approach is to separate the actual simulation and the program that implements the decision-making policy, thus enabling implementations of different heuristics to ‘‘compete’’ on the same instances and delay distributions. We also report on computational results indicating the worthiness of developing intelligent wait policies. For RDM and other logistic planning processes, it is our goal to bridge the gap between theoretical models, which are accessible to theoretical analysis, but are often too far away from practice, and the methods which are used in practice today, whose performance is almost impossible to measure.

Robust and Adaptive Network Flows

We study network flow problems in an uncertain environment from the viewpoint of robust optimization. In contrast to previous work, we consider the case that the network parameters (e.g., capacities) are known and deterministic, but the network structure (e.g., nodes and arcs) is subject to uncertainty. In this paper, we study the robust and adaptive versions of the maximum flow problem and minimum cut problems in networks with node and arc failures, and establish structural and computational results. The adaptive two-stage model adjusts the solution after the realization of the failures in the network. This leads to a more flexible model and yields less conservative solutions compared to the robust model.We show that the robust maximum flow problem can be solved in polynomial time, but the robust minimum cut problem is NP-hard. We also prove that the adaptive versions are NP-hard. We further characterize the adaptive model as a two-person zero-sum game and prove the existence of an equilibrium in such games.Moreover, we consider a path-based formulation of flows in contrast to the more commonly used arc-based version of flows. This leads to a different model of robustness for maximum flows. We analyze this problem as well and develop a simple linear optimization model to obtain approximate solutions. Furthermore, we introduce the concept of adaptive maximum flows over time in networks with transit times on the arcs. Unlike the deterministic case, we show that this problem is NP-hard on series-parallel graphs even for the case that only one arc is allowed to fail. Finally, we propose heuristics based on linear optimization models that exhibit strong computational performance for large-scale instances.

A Constant-Approximate Feasibility Test for Multiprocessor Real-Time Scheduling

We devise the first constant-approximate feasibility test for sporadic multiprocessor real-time scheduling. We give an algorithm that, given a task system and i¾?> 0, correctly decides either that the task system can be scheduled using the earliest deadline first algorithm on mspeed-(2 i¾? 1/m+ i¾?) machines, or that the system is infeasible for mspeed-1 machines. The running time of the algorithm is polynomial in the size of the task system and 1/i¾?. We also provide an improved bound trading off speed for additional machines. Our analysis relies on a new concept for counting the workload of an interval, that might also turn useful for analyzing other types of task systems.

The Price of Anarchy of a Network Creation Game with Exponential Payoff

We analyze a graph process (or network creation game) where the vertices as players can establish mutual relations between each other at a fixed price. Each vertex receives income from every other vertex, exponentially decreasing with their distance. To establish an edge, both players have to make a consent acting selfishly. This process has originially been proposed in economics to analyse social networks of cooperation. Though the exponential payoff is a desirable principle to model the benefit of distributed systems, it has so far been an obstacle for analysis. We show that the process has a positive probability to cycle. We reduce the creation rule with payoff functions to graph theoretic criteria. Moreover, these criteria can be evaluated locally. This allows us to thoroughly reveal the structure of all stable states. In addition, the question for the price of anarchy can be reduced to counting the maximum number of edges of a stable graph. This together with a probabilistic argument allows to determine the price of anarchy exactly.

Increasing Speed Scheduling and Flow Scheduling

Network flows and scheduling have been studied intensely, but mostly separately. In many applications a joint optimization model for routing and scheduling is desireable. Therefore, we study flows over time with a demand split into jobs. Our objective is to minimize the weighted sum of completion times of these jobs. This is closely related to preemptive scheduling on a single machine with a processing speed increasing over time. For both, flow scheduling and increasing speed scheduling, we provide an EPTAS. Without release dates we can prove a tight approximation factor of \((\sqrt{3}+1)/2\) for Smith’s rule, by fully characterizing the worst case instances. We give exact algorithms for some special cases and a dynamic program for speed functions with a constant number of speeds. We can prove a competitive ratio of 2 for the online version. We also study the class of blind algorithms, i.e., those which schedule without knowledge of the speed function.

Delay-Robust Event Scheduling

Robust optimisation is a well-established concept to deal with uncertainty. In particular, recovery-robust models are suitable for real-world contexts, where a certain amount of recovery---although limited---is often available. In this paper we describe a general framework to optimise event-based problems against delay propagation. We also present a real-world application to train platforming in the Italian railways in order to show the practical effectiveness of our framework.

Optimization over integers with robustness in cost and few constraints

We consider robust counterparts of integer programs and combinatorial optimization problems (summarized as integer problems in the following), i.e., seek solutions that stay feasible if at most Γ-many parameters change within a given range. While there is an elaborate machinery for continuous robust optimization problems, results on robust integer problems are still rare and hardly general. We show several optimization and approximation results for the robust (with respect to cost, or few constraints) counterpart of an integer problem under the condition that one can optimize or approximate the original integer problem with respect to a piecewise linear objective (respectively piecewise linear constraints). For example, if there is a ρ-approximation for a minimization problem with non-negative costs and non-negative and bounded variables for piecewise linear objectives, then the cost robust counterpart can be ρ(1+e)-approximated. We demonstrate the applicability of our approach on two classes of integer programs, namely, totally unimodular integer programs and integer programs with two variables per inequality. Further, for combinatorial optimization problems our method yields polynomial time approximations and pseudopolynomial, exact algorithms for Robust Unbounded Knapsack Problems.